Four-Valent Oriented Graphs of Biquasiprimitive Type

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma,G)$ where $\Gamma$ is 4-valent, connected and $G$-oriented ($G$-half-arc-transitive). Using a novel application of the structure theorem for biquasiprimitive permutation groups of the second author, we produce a description of all pairs $(\Gamma, G) \in\mathcal{OG}(4)$ for which every nontrivial normal subgroup of $G$ has at most two orbits on the vertices of $\Gamma$. In particular we show that $G$ has a unique minimal normal subgroup $N$ and that $N \cong T^k$ for a simple group $T$ and $k\in \{1,2,4,8\}$. This provides a crucial step towards a general description of the long-studied family $\mathcal{OG}(4)$ in terms of a normal quotient reduction. We also give several methods for constructing pairs $(\Gamma, G)$ of this type and provide many new infinite families of examples, covering each of the possible structures of the normal subgroup $N$.


Introduction
All graphs considered in this paper are simple, undirected and finite. A graph Γ is said to be G-oriented with respect to some group G Aut(Γ), and some edge-orientation ∆, if G acts transitively on the vertices and edges of Γ and G preserves the edgeorientation ∆. Thus, although the group G will act transitively on the vertices and edges, it will not be transitive on the arcs of Γ, where an arc is an ordered pair of adjacent vertices. Conversely, any graph Γ admitting a vertex-and edge-but not arctransitive group G of automorphisms will admit some G-invariant edge-orientation: simply take one of the two G-arc-orbits ∆ and orient an edge {α, β} from α to β if and only if (α, β) ∈ ∆.
Every G-oriented graph necessarily has even valency and all connected components of a G-oriented graph are pairwise isomorphic. It is thus natural to restrict attention to G-oriented graphs which are connected. For each even integer m 2, we let OG(m) denote the family of graph-group pairs (Γ, G) where Γ is connected, m-valent and G-oriented. Throughout this paper we will represent G-oriented graphs as pairs (Γ, G) contained in some family OG(m). Our notation suppresses the orientation ∆. Technically therefore the pairs (Γ, G) ∈ OG(m) are G-orientable rather than Goriented, as the orientation ∆ has not been chosen yet. However this orientation is determined by the pair (Γ, G) up to possibly reversing the orientation of all edges. The term G-oriented graph was suggested by B. D. Mckay and has been used, for example in [1,2,3,20] for pairs in OG(m) with an understanding that this choice The first part of this paper establishes that cases (a)-(c) of Theorem 1.1 must hold in two steps. In Section 3 we show that if (Γ, G) ∈ OG(4) is basic of biquasiprimitive type then G has a unique minimal normal subgroup N and one of the three cases (a), (b) or (c) holds for some k 1. For this we use the structure theorem for biquasiprimitive groups given in [24]. Then in Section 4 we use combinatorial arguments to obtain the various possibilities for the value of k (the number of simple direct factors of soc(G)) in each case.
In the second part of this paper (Section 5), we provide two detailed methods for constructing basic biquasiprimitive pairs (Section 5.1), and we also discuss the possibility of a third method for doing this which uses the separated box product of two digraphs as studied in [19], and which produces some, but not all, examples (Section 5.2). We conclude the paper by providing an infinite family of basic pairs for each of the cases described in Theorem 1.1, and for each possible value of k. This proves the final assertion of Theorem 1.1.

Preliminaries
Unless otherwise stated we will let V Γ, EΓ and AΓ denote the vertex-, edge-, and arc-set of a given graph Γ (an arc is an ordered pair of adjacent vertices). Given a vertex α ∈ V Γ we let Γ(α) denote the neighbourhood of α in Γ. For fundamental graph-theoretic concepts we refer the reader to [7], and for group-theoretic concepts not defined here, please refer to [25].
Given a group G acting on a set X, we will always let G X denote the subgroup of Sym(X) induced by the group G. Given elements g ∈ G and x ∈ X, we let x g denote the image of x under g. A permutation group G X is said to be semiregular if only the identity element of G fixes a point in X, and is said to be regular if it is semiregular and transitive.
2.1. G-oriented graphs. If Γ is a G-oriented graph then the group G is transitive on the vertices and edges but not on the arcs of Γ. It follows that the group G has two orbits on the arc set of Γ and these two orbits are paired. (Every arc (u, v) in one orbit will have its reverse arc (v, u) in the other orbit.) Either of these two G-orbits on the arc set of Γ naturally gives rise to a G-invariant orientation of the edges of Γ: simply take any arc (u, v) of Γ and then orient each edge {x, y} from x to y if and only if (u, v) g = (x, y) for some g ∈ G.
Given a pair (Γ, G) ∈ OG(4), any vertex v 0 ∈ V Γ and any G-invariant orientation of EΓ, we will say that an in-neighbour of v 0 is any neighbour u for which the edge {v 0 , u} is oriented from u to v 0 , and an out-neighbour of v 0 is any neighbour w for which the edge {v 0 , w} is oriented from v 0 to w. The vertex v 0 will always have exactly two in-neighbours and two out-neighbours, and the stabiliser G v0 of the vertex v 0 will always have two orbits of length two on the neighbourhood of v 0 corresponding to the in-neighbours and out-neighbours of v 0 with respect to the given orientation.
Given a connected, 4-valent, G-vertex-transitive graph Γ, we may show that (Γ, G) ∈ OG(4) by showing that G v0 has two orbits of size 2 on Γ(α), and that no element of G can reverse an edge of Γ.
An oriented s-arc of a G-oriented graph with a fixed G-invariant orientation is a sequence of vertices (v 0 , v 1 , . . . , v s ) of Γ, such that for each i ∈ {0, . . . , s − 1}, v i and v i+1 are adjacent, and each edge {v i , v i+1 } is oriented from v i to v i+1 . We will make use of the following important fact concerning oriented s-arcs of G-oriented graphs, the proof of which can be found in the first part of the proof of [3, Lemma 6.2].
Lemma 2.1. Let (Γ, G) ∈ OG(4) and let s 1 be the largest integer such that G acts transitively on the oriented s-arcs of Γ. Then G acts regularly on the oriented s-arcs of Γ. Now let (Γ, G) ∈ OG(4) and take a vertex α ∈ V Γ. Let s be as in the statement of Lemma 2.1 and consider an oriented s-arc (α, v 1 , . . . , v s ) of Γ. Since G is regular on the oriented s-arcs of Γ, it follows that the vertex-stabiliser G α is regular on the oriented s-arcs starting at α. From this it follows that G α has order 2 s , and for each i with 0 i < s, the subgroup G α,v1,...,vs−i has order 2 i . In particular, |G α,v1,...,vs−1 | = 2, and the stabiliser of a vertex G α is a 2-group.
Note that the vertex stabilisers of pairs (Γ, G) ∈ OG (4) have been studied in several papers. See for instance [13,18].
2.2. Normal Quotients. Given a pair (Γ, G) ∈ OG(4) and a normal subgroup N of G, we define a new graph Γ N called a G-normal-quotient of Γ. The vertices of Γ N are the N -orbits on the vertices of Γ, with an edge between two distinct N -orbits {B, C} in Γ N if and only if there is an edge of the form {α, β} in Γ, with α ∈ B and β ∈ C. The group G induces a group G N = G/K of automorphisms of Γ N , where K is the kernel of the G-action on Γ N . By definition N K, and hence the K-orbits are the same as the N -orbits so Γ K = Γ N . However K may be strictly larger than N .
If (Γ N , G N ) is itself a member of OG(4), that is, Γ N is a 4-valent G N -oriented graph, then Γ is said to be a G-normal cover of Γ N . In general however, the pair (Γ N , G N ) need not lie in OG(4), and the various possibilities for such normal quotient pairs (Γ N , G N ) were identified in [3, Theorem 1.1]. In particular, it was proved that for any (Γ, G) ∈ OG(4), and any nontrivial normal subgroup N of G, either (Γ N , G N ) is also in OG(4) and Γ is a G-normal cover of Γ N , or Γ N is isomorphic to K 1 , K 2 or a cycle C r , for some r 3. A pair (Γ N , G N ) where Γ N is isomorphic to one of K 1 , K 2 or C r is defined to be degenerate, while a pair (Γ, G) ∈ OG(4) for which (Γ N , G N ) is degenerate relative to every non-trivial normal subgroup N of G is defined to be basic.
Since [3, Theorem 1.1] ensures that every member of OG(4) is a normal cover of a basic pair, this result suggests a framework for studying the family OG(4) using normal quotient reduction. The goal of this framework is to improve understanding of this family by developing a theory to describe the basic pairs in OG(4), and subsequently developing a theory to describe the G-normal covers of these basic pairs. Work in this direction was initiated in [3] where the basic pairs were further divided into three types and the basic pairs of quasiprimitive type were analysed. A pair (Γ, G) ∈ OG(4) is said to be basic of quasiprimitive type if all G-normal quotients Γ N of Γ are isomorphic to K 1 . This occurs precisely when all non-trivial normal subgroups of G are transitive on the vertices of Γ. A permutation group with Algebraic Combinatorics, Vol. 4 #3 (2021) this property is said to be quasiprimitive, and there is a general structure theorem available for quasiprimitive groups analogous to the O'Nan-Scott Theorem for primitive permutation groups in [22]. Using this tool, as well as combinatorial properties of the family OG(4), it was shown [3, Theorem 1.3] that if (Γ, G) ∈ OG(4) is basic of quasiprimitive type, then G has a unique minimal normal subgroup N ∼ = T k where T is a nonabelian finite simple group and k 2.
Of course, every pair (Γ, G) ∈ OG(4) will have at least one normal quotient Γ N isomorphic to K 1 since we may take the quotient with respect to the full group G. If the only normal quotients of a pair (Γ, G) ∈ OG(4) are the graphs K 1 or K 2 , and Γ has at least one G-normal quotient isomorphic to K 2 , then (Γ, G) is basic of biquasiprimitive type. The group G here is biquasiprimitive: it is not quasiprimitive but each nontrivial normal subgroup has at most two orbits. Again, there is a structure theorem for biquasiprimitive groups available in [24].
The basic pairs in OG(4) which are neither quasiprimitive nor biquasiprimitive must have at least one normal quotient isomorphic to a cycle graph C r , and hence are said to be of cycle type. Work towards describing the basic pairs of cycle type was initiated in [2] where several important families of these graphs, which have already been discussed in the literature, were analysed from a normal quotient point of view. A more general analysis of these pairs was done in [1], however further work is required to understand this type.
The above discussion outlining the three types of basic pairs (Γ, G) ∈ OG(4) is summarised in Table 1. This table also includes references to the papers where the corresponding basic pairs were previously studied. The objective of this paper is to describe the basic pairs (Γ, G) ∈ OG(4) of biquasiprimitive type, several families of which were constructed in [20].
Before proceeding, we note that G-oriented graphs may also be divided into two types depending on the action of their full automorphism group. For a pair (Γ, G) ∈ OG(m), the graph Γ is said to be half-arc-transitive if (Γ, Aut(Γ)) ∈ OG(m). If this is not the case then Aut(Γ) preserves neither of the G-orientations of Γ, and therefore Aut(Γ) is transitive on the arc-set of Γ, that is, Γ is arc-transitive. The 4-valent halfarc-transitive graphs form a heavily studied yet elusive subfamily of OG(4), see [12,13,16,26] for instance.
For a pair (Γ, G) ∈ OG(4) and a nondegenerate normal quotient (Γ N , G N ), the graph Γ N may be either half-arc-transitive or arc-transitive, [2, Proposition 3.1]. If Γ is arc-transitive and N Aut(Γ) (N ) is arc-transitive, then Γ N is definitely arc-transitive, see [21,Lemma 1.1]. On the other hand, by considering the census of all 4-valent G-oriented graphs of order at most 1000 (see [17]), one can check that the half-arctransitive graph HAT [168,9] has two normal quotients, one of which is the half-arctransitive graph HAT[84,1], the other being an arc-transitive 4-valent graph of order 12. These normal quotients both arise from subgroups of the cyclic normal subgroup related to G-alternating cycles described in [26,Theorem 5.2,Theorem 5.6].
Half-arc-transitivity is thus difficult to discern when studying OG(4) via normal quotients. Still, once we have a good description of all basic pairs in OG(4) it would be interesting to study which of these pairs are arc-transitive and yet are normal quotients of half-arc transitive graphs.

Bi-Cayley Graphs.
A bi-Cayley graph Γ is a graph which admits a semiregular group of automorphisms H with two orbits on the vertex set of Γ. These graphs are important for our purposes as for many of the pairs (Γ, G) ∈ OG(4) studied in this paper, the group G will have a normal subgroup N contained in G + which acts semiregularly with two orbits on V Γ. In such cases, Γ is a bi-Cayley graph and the two N -orbits coincide with the two parts of the bipartition of Γ. Every bi-Cayley graph of a group H may be constructed in the following way. Let R and L be inverse-closed subsets of H which do not contain the identity, and let S be a subset of H. Define the graph Γ = BiCay(H, R, L, S) to be the graph whose vertex set is the union of the sets H 0 = {h 0 : h ∈ H} and H 1 = {h 1 : h ∈ H} (two copies of the group H), and whose edge set is the union of the right edges {{h 0 , g 0 } : gh −1 ∈ R}, the left edges {{h 1 , g 1 } : gh −1 ∈ L}, and the spokes {{h 0 , g 1 } : gh −1 ∈ S}. Note that if Γ is connected then H is generated by R ∪ L ∪ S (however the converse does not necessarily hold). The group H then acts by right multiplication on the vertices of Γ, and this action is semiregular with two orbits H 0 and H 1 . See for instance [5,27].

Biquasiprimitive Basic Pairs: two types.
Suppose now that (Γ, G) ∈ OG(4) is a basic pair of biquasiprimitive type and recall that this implies that Γ is bipartite. Let X denote the vertex set of Γ with {∆, ∆ } the bipartition of X, and let G + be the index 2 subgroup of G fixing the two biparts ∆ and ∆ setwise. Since Γ is G-vertex-transitive it follows that G + is transitive on both ∆ and ∆ .
In this section we will begin working towards the proof of Theorem 1.1. We start with a lemma about the intransitive normal subgroups of G.
Lemma 3.1. Let (Γ, G) ∈ OG(4) be basic of biquasiprimitive type, and let X denote the vertex set of Γ. Let G + be the subgroup of G of index two with orbits ∆, ∆ (the biparts of X). Then (a) G + is faithful on ∆ (and ∆ ), and (b) any non-trivial intransitive normal subgroup N of G must have the sets ∆ and ∆ as its two orbits on X. In particular, N is contained in G + .
Proof. (a). Let K be the subgroup of G + fixing ∆ pointwise and suppose that K = 1, and hence that K acts non-trivially on ∆ . If g ∈ G\G + then K g is the pointwise stabiliser of ∆ in G + , and hence K ∩ K g = 1, so K, K g ∼ = K × K g . Now since both K and K g are normal in G + , and since g 2 ∈ G + (because |G : G + | = 2), it follows that (K × K g ) g = K × K g , and so K × K g is a normal subgroup of G contained in G + . Thus K × K g has two orbits ∆ and ∆ as (Γ, G) is basic of biquasiprimitive type. But this implies that K is transitive on ∆ , which is impossible since for any α ∈ ∆ we have K G α , and G α is not transitive on Γ(α) ⊂ ∆ . Thus part (a) holds.
(b). Since |V Γ| |{α} ∪ Γ(α)| = 5, it follows that |N | subgroup of G contained in G + , so its orbits are ∆ and ∆ , and these must also be the orbits of the intransitive normal subgroup N .
Next we introduce a convenient framework for investigating these graphs, based on the Imprimitive Wreath Embedding Theorem [25,Theorem 5.5] which identifies the vertex set X with {v i | v ∈ V, i ∈ {0, 1}}, and G with a transitive subgroup of Sym(V ) Sym(2) in its natural imprimitive action, so that ∆ = {v 0 | v ∈ V } and ∆ = {v 1 | v ∈ V }. Since G is transitive, its subgroup G + induces transitive subgroups (G + ) ∆ and (G + ) ∆ on ∆ and ∆ , each of which we identify with a transitive subgroup of Sym(V ).
Let τ ∈ Sym(V ) Sym(2) generate the top group, that is, τ : v ε → v 1−ε for each v ∈ V, ε ∈ {0, 1}, and note that τ conjugates each element (h 1 , h 2 ) ∈ Sym(V ) × Sym(V ) to its reverse (h 2 , h 1 ). For a group H, y ∈ H, and ϕ ∈ Aut(H), we denote by ι y the inner automorphism of H induced by y, that is h → y −1 hy, and by Diag Proposition 3.2. Let (Γ, G) ∈ OG(4) be basic of biquasiprimitive type, and let X denote the vertex set of Γ. Let G + be the subgroup of G of index two with orbits ∆, ∆ in X, and let H be the permutation group induced by G + on ∆. Let α ∈ ∆ and β ∈ Γ(α) ⊆ ∆ . Then, if necessary, by replacing (Γ, G) with the pair (Γ g , G g ), where Γ g ∼ = Γ has edge-set (EΓ) g , for some g ∈ Sym(X), we may take X = {v i | v ∈ V, i ∈ {0, 1}}, ∆, ∆ and α = u 0 as above, for some u ∈ V , and we may identify H with a transitive subgroup of Sym(V ), such that (a) G H Sym (2), so H = (G + ) ∆ = (G + ) ∆ ; and (b) for some y ∈ H and ϕ ∈ Aut(H) with ϕ 2 = ι y , we have G + = Diag ϕ (H × H), and G = G + , g , where g := (y, 1)τ , and β = α g = (u y ) 1 . Also Proof. The assertion in part (a), that we may choose a conjugate of G and corresponding identifications of X, ∆, ∆ , so that the transitive subgroups (G + ) ∆ and (G + ) ∆ determine the same subgroup H of Sym(V ), follows from the embedding theorem [25,Theorem 5.5]. Thus G H Sym(2) = (H × H) τ , with τ as above, and G + is a subdirect subgroup of H × H. By Lemma 3.1, G + is faithful on each of ∆ and ∆ , and hence G + is a diagonal subgroup of H × H, so G + = Diag ϕ (H × H), for some ϕ ∈ Aut(H). Since α ∈ ∆, we have α = u 0 for some u ∈ V . Also, since G α < G + < G, we have G α = G + α , and since G + is a diagonal subgroup of H × H, projection to the first coordinate induces a monomorphism G + α → H with image H u . Thus G + α ∼ = H u , and we know already that G α is a 2-group.
Since G is transitive on X, there exists g = (h 1 , h 2 )τ ∈ G such that β = α g , and G = G + , g . Set s := (1, h 2 ) ∈ H × H. Then s induces a graph isomorphism from Γ to the graph Γ s with vertex set X and arc set consisting of all pairs (v s ε , w s is an arc of Γ. Moreover (Γ s , G s ) ∈ OG(4), the group G s is equal to (Diag ϕ (H × H)) s , g s , and we have (Diag ϕ (H × H)) s = Diag ϕι h 2 (H × H) and Then g s maps α s to its out-neighbour β s in Γ s , and we have α s = α, and β s = (α g ) s = (α s ) g s = α g s = (u 0 ) (y,1)τ = (u y ) 1 . Now replace Γ, G, g, ϕ, α, β by Γ s , G s , g s , ϕι h2 , α, β s . Then all assertions are proved apart from the equality ϕ 2 = ι y , which we now prove (for the new ϕ). Since g = (y, 1)τ normalises G + = Diag ϕ (H × H), it follows that, for all h ∈ H, G + Algebraic Combinatorics, Vol. 4 #3 (2021) contains (h, h ϕ ) g = (h, h ϕ ) (y,1)τ = (h ϕ , h y ) and hence we must have h y = (h ϕ ) ϕ for all h ∈ H, that is to say, ϕ 2 = ι y . Now we apply the structure theorem from [24] for biquasiprimitive groups. It turns out that only two of the various possible structures given in Theorem 1.1 of [24] can arise as groups of automorphisms of 4-valent oriented graphs of basic biquasiprimitive type. Note that the stabiliser Proof. We examine the possibilities for the structure of G given in [24, Theorem 1.1].
Since G α is a 2-group, cases (a)(iii), (b) and (c)(ii) do not arise, and since G + is faithful on ∆, the possible cases are (a)(i) and (c)(i). Consider first case (a)(i). Since |X| > 4, it follows from [24, Lemma 3.1] that the element g = (y, 1)τ does not centralise G + . A straightforward computation shows that C G + (g) consists of all pairs (h, h ϕ ) such that h ∈ C H (ϕ). Thus ϕ is nontrivial, since g does not centralise G + . Moreover in case (a)(i), H is quasiprimitive on V and the stabiliser H u ∼ = G α is a 2-group. We now apply the O'Nan-Scott Theorem for quasiprimitive groups from [22]. This theorem tells us that if H has more than one minimal normal subgroup then the stabiliser H u is not solvable. Thus H has a unique minimal normal subgroup M = soc(H) ∼ = T k where T is a simple group and k 1. Since G + ∼ = H, it follows that N = Diag ϕ (M ×M ) ∼ = M is the unique minimal normal subgroup of G + . Moreover, by [24,Lemma 3.1], no element of G G + centralises G + , and hence G ∼ = C 2 × G + , so N is the unique minimal normal subgroup of G.
It remains to consider case (c)(i). By [24, Theorem 1.1], G again has a unique minimal normal subgroup N = soc(G), and N has the form If T is a nonabelian simple group, then each minimal normal subgroup of K = T is one of the simple direct factors of this direct decomposition, and G + permutes these simple groups transitively by conjugation. The same holds for L, and as soc(G) = K × L, it follows that K and L are the only minimal normal subgroups of G + (and are interchanged by g, noting that, for (h, h ϕ ) ∈ K, the conjugate (h, h ϕ ) g = (h ϕ , h y ) ∈ L, since ϕ 2 = ι y , and vice versa). On the other hand, if T = C p , then as an H-module, M has two composition factors isomorphic to R as H-modules. In particular, H may have other minimal normal subgroups. However, for any such subgroup S we have S ∼ = R (as groups) since there are just two composition factors and both are isomorphic to R as H-modules. Also since N is the unique minimal normal subgroup of G it follows that we would have M = S × S ϕ also.

Algebraic Combinatorics, Vol. 4 #3 (2021)
In summary if (Γ, G) ∈ OG(4) is basic and biquasiprimitive, then N := soc(G) is the unique minimal normal subgroup of G, and is contained in G + . In particular N is transitive on the two G + -orbits ∆ and ∆ + , and since G α = G + α , it follows that Using the framework of Proposition 3.2, we can specify the neighbours of α = u 0 and of α g −1 = u 1 . We denote by Γ out (γ) and Γ in (γ) the 2-subsets of out-neighbours and in-neighbours of a vertex γ, respectively. Each of these two sets is an orbit of the stabiliser G γ , and we can always choose an element of G γ that acts fixed-point-freely on Γ(γ) (whether the induced group has order 2 or 4). For the vertex α, such an element is of the form (z ϕ −1 , z) for some z ∈ (H u ) ϕ . Since we did not specify above, let us now choose the G-orientation such that the vertex Lemma 3.4. Use the notation of Proposition 3.2 (in particular that g = (y, 1)τ and Proof. As mentioned above, we assume that the vertex In particular u 1 ∈ Γ out (u 0 ) and the second vertex in this set is therefore u This completes the proof of part (a). Finally applying g −1 to {α} ∪ Γ out (α) we find that Γ out (u 1 ) consists of the vertices

Biquasiprimitive Basic Pairs: restricting the socle.
We will now show that for any biquasiprimitive basic pair (Γ, G) ∈ OG(4), the unique minimal normal subgroup N of G is a direct product of k finite simple groups where k takes one of only several possible values depending on the structure of G. We deduce these values of k by separately considering the cases when N is abelian and nonabelian.
We first consider the case where the minimal normal subgroup N = soc(G) is abelian. Since N is contained in G + , this implies that N acts transitively and hence regularly on ∆ (and ∆ ). In particular, Γ is a bi-Cayley graph over N , that is, Γ = BiCay(N, ∅, ∅, S) (as defined in Subsection 2.3), and N = C k p for some k 1. Since Γ has valency 4, the cardinality |S| = 4. Moreover, by [5, Proposition 2.1], we may assume that S contains the identity of N . We will write N additively so edges of Γ are of the form (h 0 , g 1 ) where g, h ∈ C k p and g − h ∈ S, that is, g = h + s for some s ∈ S. Note that this means that a vertex adjacent to g 1 is of the form (h ) 0 with h = g − s = h + (s − s ) for some s ∈ S. It follows that Γ is connected if and only if S − S := {s − s | s, s ∈ S} generates N , and since we are assuming that the identity 0 ∈ S, this is equivalent to requiring that S {0} generates N .

3-element subset S {0}. Hence k
3. Suppose next that k = 3. Then since k is odd, it follows from Proposition 3.3 that G + = N G α is quasiprimitive on ∆ = N . In particular since N is regular on ∆, no proper non-trivial subgroup of N is normal in G + . Since N acts trivially on itself by conjugation, this implies that conjugation by G α fixes no proper non-trivial subgroup of N . However, G α is a 2-group, and N has exactly p 2 + p + 1 subgroups of order p, which is odd. Thus some subgroup of order p must be left fixed under conjugation by G α and hence must be normal in G + , a contradiction. Therefore k 2.
If p = 2 then the number of vertices is |X| = 2p k = 2 k+1 and since |X| > 4 we must have k = 2. However |G + | = |∆|.|G α | = 2 2 .|G α | 8, and G + is a 2-group, while by Proposition 3.2, |G + | = |H| for some transitive subgroup H Sym(4). Hence G + ∼ = D 8 , but then G + has a unique minimal normal subgroup of order 2 with four orbits in X, which is a contradiction.   Since N = soc(G) is nonabelian, it follows that N is a direct product of isomorphic nonabelian simple groups T . In particular, N = T k for k 1, and in case (b), k = 2 where K = T and 1. We will now show that k divides 4 in case (a) and divides 4 in case (b). As N = soc(G), we will identify N with its group of inner automorphisms Inn(N ), and regard G as a subgroup of Aut(N ) ∼ = Aut(T ) Sym(k). The representations of elements will therefore be different from Proposition 3.3.
Let s be the largest integer such that G acts transitively on the oriented s-arcs of Γ, so s 1. By Lemma 2.1, this implies that G is regular on the oriented sarcs of Γ. Consider now an oriented s-arc (v 0 , v 1 , . . . , v s ) of Γ, and suppose that the pointwise stabiliser G v0,...,vs−1 of order 2 is generated by the element h 1 , that is, We may write the automorphisms h 1 , g ∈ G as elements of Aut(N ) ∼ = Aut(T ) Sym(k), so that h 1 = f σ and g = f τ where f, f ∈ Aut(T ) k and σ, τ ∈ Sym(k). In fact in case (b), σ, τ ∈ Sym( ) Sym(2) (since in this case the -subsets of simple direct factors of the two minimal normal subgroups of G + form a G-invariant partition of In either case, h 2 1 = 1 implies that σ 2 = 1. Now let π denote the projection map π : Aut(N ) → Sym(k), so that (h 1 )π = σ and (g)π = τ , and let P := (G + )π = (N G v0 )π = (G v0 )π. Note that P is a 2-group since G v0 is a 2-group, and moreover We claim that σ is not contained in any proper τ -invariant subgroup of P . Suppose to the contrary thatP is a proper τ -invariant subgroup of P containing σ. SinceP is τ -invariant it follows that σ τ −i ∈P for all i ∈ Z, implying that P P and hence that P =P , a contradiction.
Notice that P is a subgroup of index 1 or 2 of (G)π, and the 2-group P is transitive in case (a) or has two orbits of length in case (b), so k divides |P |, or divides |P | respectively. Hence, if |P | 4 then the result follows. Thus we may assume that |P | 8, and since P is generated by conjugates of σ this means that σ = 1, so σ has order 2. In particular, P = σ , so there exists a maximal subgroup M of P containing σ . Since P is a 2-group it follows that M is normal in P . If P = (G)π then τ ∈ P , and so M is τ -invariant, contradicting the fact that σ is not contained in any proper τ -invariant subgroup of P .
Therefore P is an index 2 subgroup of (G)π, so τ ∈ (G)π\P and τ normalises P . Since g 2 ∈ G + , we have τ 2 ∈ P . Now τ does not normalise the maximal subgroup M of P containing σ, and so M 2 := M τ −1 is a maximal subgroup of P distinct from M . Let L := Φ(P ), the Frattini subgroup of P (the intersection of all maximal subgroups of P ). In particular L M ∩ M 2 , so P/L is elementary abelian of order at least 4. Also L is τ -invariant since τ normalises P , so σ ∈ L. Setting J := L, σ , it follows that J M and J/L has order 2, and conjugation by τ −1 maps J/L to (J τ −1 )/L. However, J is normal in P since P/L is elementary abelian. In particular, since τ 2 ∈ P , conjugation by τ 2 fixes J and J/L. Therefore repeated applications of conjugation by τ simply interchange the two (possibly equal) subgroups J/L and (J τ −1 )/L of P/L and each generator σ τ −i of P , lies in either J or J τ −1 . It follows that P/L is generated by J/L and J τ −1 /L, and it follows that P/L ∼ = C 2 2 , that M = J, and M 2 = J τ −1 . Thus M = L, σ , and it follows from [9, Satz III.3.2] that M = σ . This implies that |P | = 2|M | = 4, which is a contradiction. This completes the proof.
The first assertions of Theorem 1.1 now follow directly from Proposition 3.3 together with Lemmas 4.1 and 4.2.

Constructing Biquasiprimitve Pairs
In this section we complete the proof of Theorem 1. Thus Theorem 1.1 gives a total of eight different possibilities for the structure of soc(G) of a biquasiprimitive pair (Γ, G) where the number of simple direct factors is taken into account. To complete the proof, we therefore provide eight infinite families of biquasiprimitive basic pairs corresponding to these distinct cases.
In Subsection 5.1 we will describe two methods for constructing biquasiprimitive basic pairs. In short, Method 5.1 uses the standard bi-Cayley graph construction described in Subsection 2.3, while Method 5.7 is a more general coset graph construction Algebraic Combinatorics, Vol. 4 #3 (2021) developed from Proposition 3.2. All of our constructions of biquasiprimitive pairs will use one of these two methods.
The examples constructed to complete the proof of Theorem 1.1 are given in Constructions 5.11-5.30 of this section. Table 2 shows all of these constructions along with the explicit simple group T used in each case. The "Method Used" column refers to one of the two methods developed in Subsection 5.1 for producing biquasiprimitive pairs. The construction numbers are included for easy reference.  (4) is basic of biquasiprimitive type, and the unique minimal normal subgroup N of G contained in G + is semiregular (with two orbits) on V Γ, then we can take Γ to be a bi-Cayley graph Γ := BiCay(N, ∅, ∅, S) (for some subset S of N of cardinality 4). In our constructions involving bi-Cayley graphs presented in the form Γ = BiCay(N, ∅, ∅, S) we will always use the natural labelling of the vertex set V Γ. That is, we let V Γ = N 0 ∪ N 1 consisting of two copies of the group N with each vertex labelled (n) for n ∈ N and ∈ {0, 1}.
Suppose now that Γ = BiCay(N, ∅, ∅, S) where S = S −1 . Of course, such a graph is bipartite with N 0 and N 1 forming the bipartition. In order to show that Γ is connected, it suffices to show that the vertex set N 0 lies in a single connected component of Γ, or in other words that there is a path from (1 N ) 0 to (n) 0 for any n ∈ N (vertextransitivity then ensures that this holds for N 1 also). Any such path must have even length and consist of repeated left multiplication in N by an element of S followed by an element of S −1 = S. In particular, the graph Γ is connected if S 2 = N .
Hence we have the following simple method for constructing biquasiprimitive basic pairs (Γ, G). (1) S is core-free in G, g −1 / ∈ SgS, |S : S ∩ S g | = 2, and S, g = G. Moreover, for each pair (Γ, G) ∈ OG(4) there exist S G and g ∈ G such that Γ = Cos(G, S, g) and (1) holds. Specifically, for a vertex α ∈ V Γ, take S := G α and take g to be an element of G mapping α to one of its neighbours with α g 2 = α.
We can use Proposition 3.2 on the structure of biquasiprimitive basic pairs (Γ, G) ∈ OG(4) together with the coset graph construction given above to find examples of coset graphs of biquasiprimitive type. We begin by providing a general construction which uses a permutation group H (with some prescribed properties) to produce a pair (Γ, G) where Γ is a coset graph for G, and G has an index 2 subgroup isomorphic to H. In the remainder of this section we will show that under certain conditions the pairs (Γ, G) constructed in this way are basic of biquasiprimitive type. Construction 5.3. Take a permutation group H, a proper subgroup V < H, a nonidentity element y ∈ H, and an automorphism ϕ ∈ Aut(H) such that ϕ 2 = ι y , y ϕ = y, and ϕ = ι u for any u ∈ H such that u 2 = y. Now consider the group H S 2 and define two of its subgroups G + := Diag ϕ (H ×H), and S := Diag ϕ (V × V ). Also define an element g := (y, 1)(12) ∈ H S 2 . Finally construct the graph-group pair (Γ, G) where G := G + , g H S 2 and Γ := Cos(G, S, g).
It is clear that the construction of the group G in this way corresponds to the formulation of the biquasiprimitive permutation group G given in Proposition 3.2. Notice in particular that using this construction, the pair (Γ, G) is completely determined by the choices of appropriate H, V, y and ϕ. Hence we will say that a tuple (H, V, y, ϕ) is appropriate if H, V, y and ϕ satisfy the conditions of Construction 5.3. In many of the constructions that follow, we will simply apply Construction 5.3 on an appropriate (H, V, y, ϕ) to create pairs (Γ, G). The following lemma gives a sufficient condition for (Γ, G) constructed in this way to be a member of OG(4). Proof. Let G + and S be the subgroups of G defined in the construction, and let Γ = Cos(G, S, g). Suppose that (2) holds. We will show that (Γ, G) ∈ OG(4) by showing that (1) holds also. First, since H ∼ = G + , S ∼ = V , and V is core-free in H, it follows S is core-free in G + and hence is core-free in G. Next, we will show that y / ∈ V V ϕ implies that g −1 / ∈ SgS. Notice that g −1 = (1, y −1 )(12), while for any element z ∈ SgS, z = (s, s ϕ )(y, 1)(12)(t, t ϕ ) = (syt ϕ , s ϕ t) (12) for some s, t ∈ V . Thus if g −1 = z for some z ∈ SgS, then 1 = syt ϕ and hence y ∈ V V ϕ .
For the last two conditions notice that if we take x ∈ G + then x g = (h, h ϕ ) g = (h ϕ , h y ) for some h ∈ H. In particular, for s ∈ S we have s g = (t ϕ , t y ) where t ∈ V . So s g ∈ S if and only if t ϕ ∈ V . Since V ∼ = S we get that |S : Finally, it is easy to check that g 2 = (y, y) and since y ϕ = y it follows that g 2 ∈ G + . Hence if V, y = H, then S, g 2 = Diag ϕ ( V, y × V, y ) = Diag ϕ (H × H) = G + and so S, g = G.
Hence we have an easy condition for ensuring that pairs (Γ, G) formed using Construction 5.3 are contained in OG(4). Our next goal is to provide a simple condition under which such pairs are biquasiprimitive. For u ∈ H we denote by ι u the inner automorphism of H induced by conjugation by u.

Lemma 5.5. Let (Γ, G) be a graph-group pair constructed using Construction 5.3 on an appropriate (H, V, y, ϕ). Let G + and S be as defined in that construction. Then every minimal normal subgroup of G is contained in G + . In particular, if soc(G + ) ∼ = soc(H) is a minimal normal subgroup of G then it is the unique minimal normal subgroup of G.
Proof. Notice that |G : G + | = 2 since G = G + , g , g normalises G + , and g 2 = (y, y) ∈ G + . Suppose there exists a minimal normal subgroup N of G that is not contained in G + . Then, by the minimality of N it follows that G + ∩ N = 1 and hence G = G + × N . It follows that N = x ∼ = C 2 , for some x ∈ G G + . The element x has the form x = (u −1 , (u −1 ) ϕ )g = (u −1 y, (u −1 ) ϕ )(12), for some u ∈ H. Now x centralises G + , and so, for all h ∈ H we have x(h, h ϕ ) = (h, h ϕ )x, or equivalently This holds if and only if, for all h ∈ H we have (u −1 ) ϕ h = h ϕ (u −1 ) ϕ (on equating the second entries, and noting that equality in the first entries follows from this on applying ϕ). This is equivalent to requiring h = (uhu −1 ) ϕ for all h ∈ H, that is to say, ϕ is equal to ι u . In particular (u −1 ) ϕ = u −1 so x = (u −1 y, u −1 )(12), and since x 2 = (u −1 yu −1 , u −2 y) = 1, we have y = u 2 . But by our choice of element y and automorphism ϕ in Construction 1, no such u ∈ H exists. Therefore every minimal normal subgroup of G is contained in G + and hence if soc(G + ) is a minimal normal subgroup of G, then it is the unique minimal normal subgroup of G.
The above result gives the following corollary. Proof. The vertex set of Γ is the set of right cosets of S in G. Hence there are two G +orbits, namely ∆ = {Sx : x ∈ G + } and ∆ = {Sgx : x ∈ G + }. If N = soc(G + ) ∼ = M is a minimal normal subgroup of G then N is the unique such subgroup by Lemma 5.5. Moreover, the condition H = M V implies that G + ∼ = N S so N is transitive on the two G + -orbits ∆ and ∆ , and hence G is biquasiprimitive on V Γ.
The above results now provide the following method for constructing biquasiprimitive pairs in OG(4). is biquasiprimitive (by Corollary 5.6).

5.2.
A Third Method: the separated box product. While all of our explicit constructions in Subsection 5.3 use one of the two methods in Subsection 5.1, there is a third way to construct basic biquasiprimitive pairs, namely by taking the "separated box product" of quasiprimitive basic pairs (see [19]). However, as we show below in Proposition 5.9, this third method fails to produce infinitely many of the basic biquasiprimitive pairs. Suppose that (Σ, H) ∈ OG(4) is basic of quasiprimitive type and let ∆ be the 2-valent H-arc-transitive orbital digraph obtained by taking one of the two paired orbits of H on the arc-set of Σ. As in [19,Section 3], the separated box product Γ ∆ := SBP(∆, ∆) is defined to be the digraph with vertex set V Γ ∆ = V ∆ × V ∆ × Z 2 , and arcs being all ordered pairs ((α, γ, 0), (β, γ, 1)) with (α, β) an arc of ∆ and γ ∈ V Γ, together with all ordered pairs ((β, γ, 1), (β, δ, 0)) with (γ, δ) an arc of ∆ and β ∈ V Γ. Then Γ ∆ is a (possibly disconnected) bipartite 2-valent digraph, and by [19,Corollary 3.4], the group G ∼ = H S 2 acts naturally on Γ ∆ as a subgroup of automorphisms, G is transitive on the arcs of Γ ∆ , and its index 2 subgroup H × H fixes setwise the two parts of the bipartition. A discussion of connectivity of Γ ∆ is given in [19,Section 3.3]. If Γ ∆ is connected and Γ is its underlying graph, then our discussion shows that (Γ, G) ∈ OG(4). Moreover, since (Σ, H) ∈ OG(4) is basic of quasiprimitive type, it follows from [3, Theorem 1.3] that H has a unique minimal normal subgroup, say N , and N is transitive on V Σ = V ∆. Hence G also has a unique minimal normal subgroup N × N , and N × N is transitive on each part of the bipartition of V Γ. This implies that G is biquasiprimitive on V Γ, and so (Γ, G) is basic of biquasiprimitive type. Thus we have the following method for constructing a basic pair (Γ, G) ∈ OG(4) of biquasiprimitive type.

(4) Show that Γ is connected, and if this is the case then return (Γ, G).
Our discussion shows that each pair (Γ, G) produced by Method 5.8 lies in OG(4) and is basic of biquasiprimitive type. It turns out, however, that many families of basic pairs of biquasiprimitive type cannot be constructed using Method 5.8. To explain why, we note that, by [3,Theorem 1.3], if (Σ, H) ∈ OG(4) is basic of quasiprimitive type, then H has a unique minimal normal subgroup N = T such that T is a nonabelian simple group and 2. Hence for any basic biquasiprimitive pair (Γ, G) constructed by Method 5.8, the group G will have a unique minimal normal subgroup soc(G) = T 2 with 2. To start with, we will not obtain the examples (Γ, G) we give in Constructions 5.11 and 5.14 where the group G has an abelian socle. More than this is true. Our next result shows we will not obtain the examples arising from Constructions 5. 16 and 5.30 where the socle of G is T k with k = 1 and k = 8, respectively.
Proposition 5.9. Suppose that (Γ, K) ∈ OG(4) is basic of biquasiprimitive type and that soc(K) = T k where T is a nonabelian simple group and k ∈ {1, 8}. Then the oriented graph Γ cannot be constructed using Method 5.8.
Proof. Let (Γ, K) be as in the statement of the proposition, let S := soc(K), and note that S is the unique minimal normal subgroup of K, by Theorem 1.1. Now suppose for a contradiction that Method 5.8, applied to a basic quasiprimitive pair (Σ, H) ∈ OG(4) produces a basic biquasiprimitive pair (Γ, G) ∈ OG(4), where the groups G = H S 2 and K preserve the same orientation of Γ. By the discussion preceding the proposition U := soc(G) = L 2 , where L is a nonabelian simple group and 2 ∈ {2, 4}. In particular G = K. We will show that this leads to a contradiction.
As usual, we denote by K + the index 2 subgroup of K which preserves the two parts of the bipartition of Γ. Taking a vertex α ∈ V Γ we then have, K + = SK α and K α is a 2-group, say |K α | = 2 a 2. Hence |K| = 2|K + | = 2|S|.|K α |/|S α | = 2 a |T | k for some a such that 1 a a + 1. Also |V Γ| = |K|/2 a = 2 a−a |T | k . Now let X = G, K , and note that X Aut(Γ) and X preserves the same orientation of Γ preserved by K and G. Since K = G, the group K is a proper subgroup of X, and since X α is a 2-group, it follows that |X| = |V Γ||X α | = 2 b |T | k for some b a + 1.
Let Y be the last term in the derived series for X. Then X/Y is solvable, and hence its subgroup SY /Y ∼ = S/(S ∩ Y ) is soluble, which implies that S is contained in Y . Now let N be a normal subgroup of X properly contained in Y , and suppose that N is maximal with respect to these properties. Then Y /N is a minimal normal Hence Y /N is nonabelian and thus Y /N ∼ = R c for some nonabelian simple group R and c 1.
Since the index |Y : S| is a 2-power, it follows that S is not contained in N (as otherwise Y /N would be a 2-group). Thus S ∩ N is a proper subgroup of S. Moreover, since both S and N are K-invariant, it follows that S ∩ N is normalised by K. However, since S is minimal normal in K we conclude that S ∩ N = 1. Therefore S ∼ = SN/N Y /N ∼ = R c and so S = T k is isomorphic to a subgroup of R c . Note that since S ∩ N = 1 it also follows that N ∩ K = 1, as S is the unique minimal normal subgroup of K. Now write R c = R 1 × · · · × R c , and for each i let π i : R c → R i denote the natural projection map. Notice that if π i (S) = 1 for some i, then π i (S) ∼ = T j for some j 1, and the index |R i : π i (S)| is a power of 2. Thus it follows from [8, Theorem 1] that j = 1 and either R = T ; or R = A n , T = A n−1 and |R : T | = n = 2 d for some d. In either case, since |R c |/|T | k is a power of 2, it follows that k = c.

Algebraic Combinatorics, Vol. 4 #3 (2021)
We can apply exactly these arguments with G, U, L, 2 in place of K, S, T, k, and this yields in particular that c = 2 . Thus we conclude that k = 2 which is a contradiction as either k = 1 < 2 or k = 8 > 2 .
Remark 5.10. The proof of Proposition 5.9 raises several questions about whether a nonabelian group K, where (Γ, K) ∈ OG(4) is basic of biquasiprimitive type, could be embedded into a larger group X Aut(Γ) which preserves the same orientation of Γ as K does. In particular, taking T k , N , and R c to be as defined in the proof of Proposition 5.9, the arguments there imply that k = c, N is a 2-group, and either Construction 5.11. Take a prime p ≡ 1 (mod 4) and let q ∈ Z p such that q 2 ≡ −1 mod p. Let Γ = BiCay(N, ∅, ∅, S) with vertex set N 0 ∪ N 1 , where N = Z p and S = {±1, ±q}. Define a permutation δ of the vertices of Γ by x δ ε = (x · q) 1−ε for ε ∈ {0, 1}, and set G := N δ .
Lemma 5.13. For Γ, G as in Construction 5.11, (Γ, G) ∈ OG(4) and is basic of biquasiprimitive type with soc(G) as described in Theorem 1.1 case (a) with k = 1.
Proof. Since |S| = 4 and S = S 2 = N it follows that Γ is 4-valent and connected. Also by Proposition 5.2, δ ∈ Aut(Γ) since it is induced by an automorphism of N fixing S setwise. Notice that the automorphism δ has order 4 and that the stabiliser in G of the vertex (0) 0 is δ 2 ∼ = C 2 . This group has two orbits of length two on the neighbourhood of (0) 0 , namely {(1) 1 , (−1) 1 } and {(q) 1 , (−q) 1 }. Now, any automorphism g ∈ G is of the form g = nδ i with n ∈ N and i ∈ {1..4}. In particular, any automorphism taking the vertex (0) 0 to its neighbour (1) 1 must be of the form g = nδ i with n ∈ N and i ∈ {1, 3}. This gives just two possibilities for such an automorphism namely g 1 = q 3 δ and g 2 = qδ 3 where q ∈ N . These two automorphisms map (1) 1 to (1 + q) 0 and (1 − q) 0 respectively. Thus no element of G can reverse edges and Γ is G-oriented.

Algebraic Combinatorics, Vol. 4 #3 (2021)
Since the only proper non-trivial normal subgroups of G are N and N δ 2 it follows that (Γ, G) is basic of biquasiprimitve type.
Lemma 5.15. For Γ, G as in Construction 5.14, (Γ, G) ∈ OG(4) and is basic of biquasiprimitive type with soc(G) as described in Theorem 1.1 case (a) with k = 2.
To show that (Γ, G) is basic of biquasiprimitive type, notice that the setwise stabiliser G + in G of the two parts N 0 and N 1 of V Γ is N δ 2 , with δ 2 acting as inversion on N . Hence the nontrivial normal subgroups of G + are N , and the subgroups of N isomorphic to Z p (all intransitive on N 0 since N is regular). Therefore we need to check that none of the subgroups of N of order p is normal in G.
To this end, notice that the subgroups corresponding to the direct factors of N are swapped by conjugation by δ in G, and hence aren't normal. All other nontrivial proper subgroups of N are of the form (1, It follows that c = x and so x 2 ≡ −1 mod p, but this is impossible since p ≡ 3 mod 4. Thus the only proper non-trivial normal subgroups of G are N and G + , both of which are transitive on the two biparts of V Γ. Next we give constructions of biquasiprimitive basic pairs (Γ, G) ∈ OG(4) with soc(G) nonabelian. Note that any nonabelian simple group T can be generated by an involution and an element of prime order [10]. In particular all nonabelian simple groups can be generated by two elements. In each of our constructions of biquasiprimitive pairs with nonabelian socle we will use a simple group T and a generating pair {a, b} with prescribed properties.
We begin with constructions of biquasiprimitive basic pairs (Γ, G) ∈ OG(4) with soc(G) nonabelian and as described in Theorem 1.1 case (b). Proof. Since N is nonabelian and the orders of b and ab are odd, it follows that S 0 ∩ S −1 0 = ∅ (as ab = (ab) −1 ) and hence that |S| = 4 and Γ is 4-valent. Again, using the fact that b has odd order it is easy to check that a, b ∈ S and hence that S = N .
Algebraic Combinatorics, Vol. 4 #3 (2021) Now consider S 2 . This set contains the elements abab, b 2 and baba. In particular, S 2 contains b and hence also contains aba. Since aba and abab are contained in S 2 and the order of ab is odd, it follows that a ∈ S 2 and hence S 2 = N . Therefore Γ is connected.
Next, notice that both σ and δ are induced by conjugation by a in N and this automorphism fixes S setwise. Hence σ and δ are automorphisms of Γ by Proposition 5.2. The stabiliser in G of the vertex (1 N ) 0 is σ with two orbits on the neighbours of (1 N ) 0 , namely {(ab) 1 , (ba) 1 } and {(b −1 a) 1 , (ab −1 ) 1 }. Furthermore a straightforward check shows that the only automorphisms in G mapping 1 0 to (ab) 1 are g 1 = (ab)σδ and g 2 = (ba)δ (where (ab) and (ba) are automorphisms contained in N ) and neither of these map (ab) 1 to 1 0 . This implies that Γ is G-oriented and hence that (Γ, G) ∈ OG (4). Now notice that neither σ nor δ is normal in G. On the other hand, N is a normal (and hence minimal normal) subgroup of G, and is the unique such subgroup. Since N clearly has two orbits on V Γ, it follows that G is biquasiprimitive on the vertices of Γ. To see that Γ is connected consider the following. The projections of S onto the simple direct factors of N = T × T are both equal to the group a, b = T . Hence either S = N or S = {(t, t ϕ ), t ∈ T } for some ϕ ∈ Aut(T ). In the latter case, (a, b) = (a, a ϕ ) so b = a ϕ , but also (b, a) = (b, b ϕ ) so a = b ϕ , but by our assumption no such automorphism ϕ exists. Hence N = S . Finally, notice that since both a and b have odd order, we have (a, b) ∈ (a 2 , b 2 ) (and similarly (b, a) ∈ (b 2 , a 2 ) ). In particular both (a, b) and (b, a) are contained in S 2 , so N = S = S 2 , and Γ is connected.
Once again Proposition 5.2 implies that σ, δ ∈ Aut(Γ). Now it is clear that G acts transitively on the vertices of Γ and the stabiliser in G of the vertex (1 N ) 0 is exactly σ ∼ = C 2 with two orbits on the neighbourhood of (1 N ) 0 . Moreover, it is easy to check that no automorphism can reverse edges as follows. The only automorphisms taking (1 N ) 0 to (a, b) 1 are g 1 = n 1 σδ and g 2 = n 2 δ where n 1 = (a, b) and n 2 = (b, a) are elements of N . Since neither of these maps (a, b) 1 to (1 N ) 0 , it follows that Γ is G-oriented and (Γ, G) ∈ OG(4).
Finally, since conjugation by σ in G interchanges the two simple direct factors of N , it follows that N is a minimal normal subgroup of G and so is the unique minimal normal subgroup. Of course, N has two orbits on V Γ, thus G is biquasiprimitive on the vertices of Γ.
Algebraic Combinatorics, Vol. 4 #3 (2021) Next we give a construction of biquasiprimitive basic pairs as described in Theorem 1.1 case (b) with k = 4. This time we will use Method 5.7. We will use the same simple group T and generating pair {a, b} in Constructions 5.23, 5.28 and 5.30. Hence we begin with the following important remark. In order to apply this method we must first show that the quadruple (H, V, y, ϕ) given in these constructions is appropriate. In each of these constructions, the fact that the automorphism ϕ 2 = ι y and the fact that y ϕ = y follow immediately from the choices of ϕ ∈ Aut(H) and y ∈ H. Thus showing that (H, V, y, ϕ) is appropriate amounts to showing that ϕ = ι u for any u ∈ H with u 2 = y.
In each of these three constructions the group H is of the form T k V T k S k , where k 1 and V is an elementary abelian 2-group. We may thus define two types of projection maps as follows. Let ρ be the projection map ρ : T S k → S k and, for each i ∈ [1, k], let π i be the projection map π i : T 1 × · · · T k → T i . Now if ϕ = ι u for some u ∈ H with y = u 2 , then we have u = ts where t ∈ T k , s ∈ ρ(H) and s 2 = 1. Furthermore, since u 2 = y it follows that u = ts and y commute, which implies that y ts = y and hence that y t = y s . We then have π s(i) (y) = π i (y s ) = π i (y t ) = π i (y) πi(t) and it follows that, if s interchanges the ith and jth simple direct factors of T k , then π j (y) and π i (y) lie in the same conjugacy class of T , a fact which we will use in proving that each of these constructions gives appropriate (H, V, y, ϕ). Proof. We begin by showing that the quadruple (H, V, y, ϕ) as given in Construction 5.23 is appropriate. By Remark 5.24 we only need to show that ϕ = ι u for any u ∈ H with u 2 = y. Suppose then, for a contradiction, that ϕ = ι u for some u ∈ H such that u 2 = y. By Remark 5.24 we know that u = ts for some t ∈ T 4 and s ∈ ρ(H) = ρ(V ) where, if s interchanges i and j, then π i (y) and π j (y) lie in Algebraic Combinatorics, Vol. 4 #3 (2021) the same conjugacy class of T . Consider the order |π (y)| for each . It is easy to check that (|bab|, |baba|, |ab 2 |, |ab 2 a|) = (p, p, p, 3), and hence that ab 2 a cannot lie in the same conjugacy class as any of the other entries of y. On the other hand, since each nontrivial element c ∈ ρ(H) interchanges two pairs of elements of {1, 2, 3, 4}, it follows that s = 1. Thus ϕ = ι u for some u ∈ T 4 . However, it is clear that if we take z : = (1, 1, a, 1) ∈ H, then z ϕ = z u for any u ∈ T 4 since π 1 (z u ) = 1 for each u ∈ T 4 , while π 1 (z ϕ ) = a ab = 1. Thus ϕ = ι u for any u ∈ T 4 , a contradiction. Hence by Remark 5.24, the quadruple (H, V, y, ϕ) as given in Construction 5.23 is appropriate.
Thus Construction 5.23 is a special case of Construction 5.3. Hence, in order to show that (Γ, G) ∈ OG(4) it suffices to show that condition (2) of Lemma 5.4 is satisfied. First notice that V ∼ = Z 2 2 since h 1 and h 2 are commuting involutions. Also V is core-free in H since for instance V ∩ V y = 1. It is also easy to check that V ∩ V ϕ = h 2 , and so |V : Now suppose that y ∈ V V ϕ so that y = vu for some v ∈ V, and u ∈ V ϕ . This implies that vu ∈ T 4 , and hence, if we again take ρ to be the projection map T S 4 → S 4 , then ρ(v) = ρ(u). Hence the only possibilities for (v, u) such that y = vu that need to be considered are (h 1 , h ϕ 2 ), (h 2 , h 2 ), and (h 1 h 2 , h 2 h ϕ 2 ). The second possibility gives h 2 2 = 1 = y, while the first and third possibilities both give y = h 1 h ϕ 2 . It is easy to check however that h 1 h ϕ 2 = h 1 h y 1 has bab 2 in its third coordinate while y has ab 2 in its third coordinate. Hence y / ∈ V V ϕ . It remains to show that V, y = H, and to prove this it is sufficient to show that T 4 V, y . To this end, let y 1 := y h1 and y 2 := y h2 , so that we have y = (bab, baba, ab 2 , ab 2 a), , and y 2 = (b 2 ab 2 ab, ab 2 abababa, b 2 abab 2 , ab 2 ).
We claim that T 4 = y, y 1 , y 2 V, y . First, it is straightforward to check that the group y, y 1 , y 2 projects onto each simple direct factor of T 4 . Consider now the elements of T appearing as coordinates of y, y 1 and y 2 . It is easy to see that the three elements ab 2 a, b 2 , and b 2 ab 2 ab have order 3. On the other hand, using the fact that ab and ab 2 have order p, we can check that abab, bab and b 2 abab 2 also have order p. The remaining elements appearing as coordinates of y, y 1 and y 2 are conjugates of these elements of order p and hence also have the same order. In particular, since the only elements of order 3 (ab 2 a, b 2 , and b 2 ab 2 ab), appear in the fourth, third and first coordinates of y, y 1 and y 2 respectively, and y, y 1 , y 2 is a subdirect subgroup of T 4 , it follows that T 4 = y, y 1 , y 2 and so V, y = H. Hence, by Lemma 5.4, (Γ, G) ∈ OG(4).
Finally we show that (Γ, G) is basic of biquasiprimitive type. Since H acts transitively on the simple direct factors of T 4 , it follows that T 4 is a minimal normal subgroup of H, and is the unique such subgroup. Hence N = Diag ϕ (T 4 × T 4 ) ∼ = T 4 is the unique minimal normal subgroup of G + , and must be the unique minimal normal subgroup of G. Hence (Γ, G) is biquasiprimitive by Corollary 5.6.
We conclude this section by giving constructions of basic biquasiprimitive (Γ, G) ∈ OG(4) as described in Theorem 1.1 case (c). The first construction is similar to Construction 5. 19. As in that construction, the alternating group Alt(n) with n odd, and generators a = (123) and b = (12 . . . n) will have the required properties. Proof. Since Γ is the same graph from Construction 5.19, it follows from Lemma 5.21 that Γ is 4-valent and connected. Again σ and δ are induced by automorphisms of N which fix S and hence are automorphisms of Γ by Proposition 5.2. Moreover it is a straightforward check that the stabiliser in G of the vertex (1 N ) 0 is σ ∼ = C 2 and also that there are only two automorphisms in G mapping the vertex (1 N ) 0 to its neighbour (a, b) 1 but neither of these reverses the edge {(1 N ) 0 , (a, b) 1 }. Hence Γ is G-oriented and (Γ, G) ∈ OG(4). Now notice the setwise stabiliser in G of ∆ := N 0 is G + = N σ and that T ×1 N is a normal subgroup of G + which is intransitive on ∆. In particular, G + is not quasiprimitve on ∆. Moreover δ interchanges the two simple direct factors of N , and hence N is the unique minimal normal subgroup of G. Since N is contained in G + it follows that Γ is basic of biquasiprimitive type as in Lemma 4.2 case (b).
The next two constructions both provide pairs (Γ, G) as described in Theorem 1.1 case (c) with = 2 and = 4 respectively. In both cases soc(G) = T 2 where T is the simple group PSL (2, p). In both cases we may use the same generating pairs {a, b} as those used in Construction 5.23 (see Remark 5.22). Proof. We begin by showing that the quadruple (H, V, y, ϕ) as given in Construction 5.28 is appropriate. By Remark 5.24 we only need to show that ϕ = ι u for any u ∈ H with u 2 = y. Suppose then, for a contradiction, that ϕ = ι u for some u ∈ H such that u 2 = y. By Remark 5.24 we know that u = ts for some t ∈ T 4 and s ∈ ρ(H) = ρ(V ), where if s interchanges i and j then π i (y) and π j (y) lie in the same conjugacy class of T . Now consider the order |π (y)| for each . It is easy to check that (|b 2 a|, |ab 2 a|, |bab|, |b 2 |) = (p, 3, p, 3), implying that s cannot interchange 1 and 2. However, since s ∈ ρ(H) = (12)(34) it follows that s = 1, and so ϕ = ι u for some u ∈ T 4 . It is easy to see that, if we take z := (1, 1, a, 1) ∈ H then z ϕ = z u for any u ∈ T 4 , since π 1 (z u ) = 1 for all u ∈ T 4 while π 1 (z ϕ ) = a b 2 = 1. This contradiction implies that ϕ = ι u for any u ∈ H with u 2 = y. Hence by Remark 5.24, the quadruple (H, V, y, ϕ) as given in Construction 5.28 is appropriate.
Algebraic Combinatorics, Vol. 4 #3 (2021) Thus Construction 5.28 is a special case of Construction 5.3. Hence, in order to show that (Γ, G) ∈ OG(4) it is sufficient to show that condition (2) of Lemma 5.4 is satisfied. Here V ∼ = C 2 and since h ϕ 1 / ∈ V we have that V is core-free in H and |V : V ∩ V ϕ | = 2. It is also easy to check that y / ∈ V V ϕ in this case, by noticing that y = h 1 h ϕ 1 . It remains to show that V, y = H. In fact, we will show that T 4 y 1 , y where y 1 := y h1 = (b 2 , ab 2 , ab 2 a, ababa), from which it follows that V, y = H. First, y, y 1 projects onto each simple direct factor of T 4 , so we only need to make sure that y, y 1 is not a product of diagonal subgroups of T 4 . Now y has elements of order p in its first and third coordinates and elements of order 3 in its second and fourth coordinates. Thus all we need to check is that no automorphism of T can map b 2 a to bab and b 2 to ab 2 a and that no automorphism can map ab 2 a to b 2 and ab 2 to ababa. In the first case, such an automorphism must map a to (ab) 3 which is impossible since a is an involution. In the second case, such an automorphism must map a to (ab 2 ) 3 , which again is not an involution. Hence T 4 y 1 , y and so V, y = H. By Lemma 5.4 we conclude that (Γ, G) ∈ OG(4).
It is clear that the action of H on the simple direct factors of T 4 has two orbits of length 2. Thus H has two minimal normal subgroups isomorphic to T 2 , and these are the only minimal normal subgroups of H. Furthermore, it is clear that the automorphism ϕ of H interchanges these normal subgroups. Let R and R ϕ denote these two minimal normal subgroups of H.
Since G + ∼ = H, G + also has two minimal normal subgroups isomorphic to R, R ϕ ∼ = T 2 . Let K and L denote these minimal normal subgroups of G + so K = Diag ϕ (R×R) and L = Diag ϕ (R ϕ × R ϕ ). Then conjugation by g in G, interchanges K and L and so G acts transitively on the direct factors of soc(G + ) = K × L ∼ = T 4 . Hence soc(G + ) is a minimal normal subgroup of G and (Γ, G) is biquasiprimitive by Corollary 5.6.  = (a, a, a, a, a, a, a, a) Proof. We begin by showing that the quadruple (H, V, y, ϕ) as given in Construction 5.30 is appropriate. By Remark 5.24 we only need to show that ϕ = ι u for any u ∈ H with u 2 = y. Suppose then, for a contradiction, that ϕ = ι u for some u ∈ H such that u 2 = y. By Remark 5.24 we know that u = ts for some t ∈ T 8 and s ∈ ρ(H) = ρ(V ), where if s interchanges i and j then π i (y) and π j (y) lie in the same conjugacy class of T . Now consider the order |π (y)| for each . It is easy to check that (|1|, |a|, |ab 2 a|, |ab 2 |, |1|, |ababa|, |b 2 |, |ab 2 aba|) = (1, 2, 3, p, 1, p, 3, 2).