Cayley diagrams of groups of genus 1

The pink arrows around the edge of each diagram are "sewing instructions", showing how it is to be assembled into a genus 1 surface, or torus. The light pink regions are to be "discarded" once the sewing has been done.

A4
A4
A4
A4
A4×C2
A4×C2
A4×C2
A4×C2
SL(2,3)
SL(2,3)
S4
S4
(C3×C3)⋊C3
(C3×C3)⋊C3
(C3×C3)⋊C2
(C3×C3)⋊C2
(C2×C2)⋊C4
(C2×C2)⋊C4
quasidihedral, order 16
quasidihedral 16
modular, order 16
modular 16
pauli
Pauli
pauli
Pauli
C4⋊C4
C4⋊C4
C4×C7, as an example of the many groups of the form Cm×Cn
C4×C7
Q8
Q8
Q8
Q8
Q28, as an example of the many dicyclic groups
Q28
C3⋊C8
C3⋊C8
C3⋊D8
C3⋊D8
C2×D14, as an example of the many groups of the form C2×D2n
C2×D14n
C2×C2×C2×C2
C2×C2×C2×C2
Q8×C3
Q8×C3
C7×C2×C2, as an example of the many groups of the form C2×C2×Cn
C7×C2×C2
C5×D6, as an example of the many groups of the form Cn×D2n
C5×D6
(C2×C2)⋊C4
(C2×C2)⋊C4
C5⋊C4, the Frobenius group of order 20
C5⋊C4, Frobenius 20

(C3xC3)⋊C2
(C3xC3)⋊C2
C7⋊C3
C7⋊C3, Frobenius 21
C7⋊C6, the Frobenius group of order 42
C7⋊C6, Frobenius 42

Dicyclic groups

These need redrawing, they are here portrayed on a bounded infinite strip of the plane, with the arcs crossing.

The groups portrayed are specified to the left. Any finite dicyclic group can be portrayed in the same ways.

Qℤ 
Qℤ 
Qℤ 

Regular maps drawn on the torus.
Some more Cayley diagrams drawn on surfaces appropriate to their genus.
Some more Cayley diagrams
and other pages on groups
Copyright N.S.Wedd 2009