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**.

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 | M_{12} | sharply 5 | 12 | |

dual of Klein map | PSL(2,7) | <2,4 ; 4,3> | 12 | S4 | M_{24} | 5 | 24 |

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 |

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>****<m,n>****m**th edge from that vertex,
counting from the left; continue to the next vertex; select the **n**th 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>****<1,2>**

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

A solidus indicates alternation. Thus **<5,4 / 4,3>****S**. From a member of the first class use
**<5,4>****<4,3>****S** which is not as defined above,
not being half-edge transitive.

More on Regular Maps

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