D8

Also called C4 ⋊ C2.

Statistics
Order of group	8
GAP identifier	8,3
Presentation	< k,r \| r⁴, r², (kr)² >
Orders of elements	1 of 1, 1+2*2 of 2, 2 of 4
Centre	C2
Derived subgroup	C2
Automorphism group	D8
Inner automorphism group	C2×C2
"Out" (quotient of above)	C2
Schur multiplier	C2

Permutation Diagrams

1-transitive on 4
points, odd.

1-transitive on 4
points, odd.

1-transitive on 4
points, odd.

1-transitive on 4
points, odd.

1-transitive on 8
points, even.

Cayley Graphs

the cube, type I

the di-square, type II

the cube, type I

Regular maps with D8 symmetry

D8 is the rotational symmetry group of the regular maps the 4-hosohedron, the di-square, the hemi-4-hosohedron, the hemi-di-square, the 4-lucanicohedron, the hemi-4-lucanicohedron.

D8 is the full symmetry group of the regular map {4,4}_(1,0).

Index to regular maps