D20

Also called  D10×C2.

Statistics

Order of group20
GAP identifier20,4
Presentation< k,r | k10, r2, (kr)2 >
Orders of elements1 of 1, 1+2*5 of 2, 2*2 of 5, 2*2 of 10
CentreC2
Derived subgroupC5
Automorphism groupFrob20×C2
Inner automorphism groupD10
"Out" (quotient of above)C2×C2
Schur multiplierC2
Sylow-2-subgroupC2×C2
 

Permutation Diagrams


Not transitive.

Not transitive.

1-transitive on 10
points, odd.

1-transitive on 10
points, odd.

1-transitive on 10
points, odd.

Not transitive.

1-transitive on 20
points, even.

Cayley Graphs



the 10-hosohedron, type II


the 5-hosohedron, type IIIa


the 5-hosohedron, type III




Regular maps with D20 symmetry

D20 is the rotational symmetry group of the regular maps the 10-hosohedron,   the di-decagon,   the hemi-10-hosohedron,   the hemi-di-decagon,   the 10-lucanicohedron,   the hemi-10-lucanicohedron.

D20 is the full symmetry group of the regular maps S2:{10,5},   S2:{5,10},   the 5-hosohedron,   the di-pentagon,   the 5-lucanicohedron,   rectification of S2:{10,5}.


Index to regular maps

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