It is known that there are precisely three transitive permutation groups of degree 6 that admit an invariant partition with three parts of size 2 such that the kernel of the action on the parts has ...order 4; these groups are called
A
4
(
6
),
S
4
(
6
d
) and
S
4
(
6
c
). For each
L
∈
{
A
4
(
6
)
,
S
4
(
6
d
)
,
S
4
(
6
c
)
}, we construct an infinite family of finite connected 6‐valent graphs
{
Γ
n
}
n
∈
N and arc‐transitive groups
G
n
≤
Aut
(
Γ
n
) such that the permutation group induced by the action of the vertex‐stabiliser
(
G
n
)
v on the neighbourhood of a vertex
v is permutation isomorphic to
L, and such that
∣
(
G
n
)
v
∣ is exponential in
∣
V
(
Γ
n
)
∣. These three groups were the only transitive permutation groups of degree at most 7 for which the existence of such a family was undecided. In the process, we construct an infinite family of cubic 2‐arc‐transitive graphs such that the dimension of the 1‐eigenspace over the field of order 2 of the adjacency matrix of the graph grows linearly with the order of the graph.
PDF