Regular maps and multiply-transitive groups

This page describes a way to derive multiply-transitive groups from regular maps.

It is most likely to work if the regular map that we start with

has triangular faces,
has an even number of vertices,
has a vertex multiplicity of 1 (i.e. no two edges connect the same pair of vertices)

The process is as follow.

Take a regular map R, preferably satisfying most of the above conditions
Find a subset of edges of R, we will call it S, such that the rotational symmetry group of S acts half-edge-transitively on S, and such that no two edges of S share a vertex. Note that by Lagrange's theorem, [rsg(S)] must divide [rsg(R)], and so the number of edges of S must divide the number of edges of R.
Generally, try to choose S as large as possible, at or close to its upper limit of half the number of vertices of R. (If the result is to be sharply 3-transitive, at most 2 vertices can remain fixed by any element.)
Find the group, permuting the vertices of R, that is generated by the rotational symmetry group of R, plus the involution which interchanges the vertices at the ends of each edge in S.

Examples

Regular map R	rsg(R)	Instruction set	size of S	rsg(S)	R–S	Group obtained	degree of transitivity	Size of set permuted
Octahedron	S4	<1,2>	3	D6	Triangular prism	S5	sharply 3	6
Cube	S4	<1,2>	3	D6	3-hosohedron	PGL(2,7)	sharply 3	8
Icosahedron	A5	<4,1>	5	D10	pentagonal prism capped with pyramids	PGL(2,11)	sharply 3	12
dual of Dyck map	(96 elements)	<5,4 / 4,3>	6	C3		M₁₂	sharply 5	12
dual of Klein map	PSL(2,7)	<2,4 ; 4,3>	12	S4		M₂₄	5	24

In the table above,

rsg		denotes rotational symmetry group
instruction set		is explained below
R–S		denotes the (usually) non-regular map formed by removing the edges of S from R

Instruction sets

It is useful to be able to specify a particular set S. We do this by an "instruction set" which shows how to get from one of S's member edges to another. A typical instruction set looks like this: <1,3>. An instruction set <m,n> means "travel along the edge you have selected until you reach a vertex; select the mth edge from that vertex, counting from the left; continue to the next vertex; select the nth edge from that vertex, counting from the left."

The illustration shows an icosahedron, with a set S in yellow (and five other conjugate sets shown in five other colours). This S could be specified as <4,1> or as <1,2>.

A semicolon indicates a choice. Thus <2,4 ; 4,3> means "take the 2nd edge then the 4th edge, or take the 4th edge then the 3rd edge".

A solidus indicates alternation. Thus <5,4 / 4,3> means "there are two classes of edge in S. From a member of the first class use <5,4> to reach a member of the second class. From a member of the second class use <4,3> to reach a member of the first class." This results in an S which is not as defined above, not being half-edge transitive.