D12

Also called  D6 x C2.

Statistics

Order of group12
GAP identifier12,4
Presentation< k,r | k6, r2, (kr)2 >
Orders of elements1 of 1, 1+2*3 of 2, 2 of 3, 2 of 6
CentreC2
Derived subgroupC3
Automorphism groupD12
Inner automorphism groupD6
"Out" (quotient of above)C2
Schur multiplierC2
Sylow-2-subgroupC2×C2
 

Permutation Diagrams


Not transitive.

Not transitive.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

Not transitive.

1-transitive on 12
points.

Cayley Graphs


the 6-hosohedron, type II

the di-dodecagon, type I


the 3-hosohedron, type III

the 3-hosohedron, type IIIa





Regular maps with D12 symmetry

D12 is the rotational symmetry group of the regular maps the 6-hosohedron,   the di-hexagon,   the hemi-6-hosohedron,   the hemi-di-hexagon,   the 6-lucanicohedron,   the hemi-6-lucanicohedron.

D12 is the full symmetry group of the regular maps {3,6}(1,1),   {6,3}(1,1),   the 3-hosohedron,   the di-triangle,   the 3-lucanicohedron,   rectification of {6,3}(1,1).


Index to regular maps

Orientable sphere | torus | 2 | 3 | 4 | 5 | 6
Non-orientable projective plane | 4 | 5 | 6 | 7