A4

Also called  PSL(2,3),   tetrahedral group,   (C2×C2) ⋊ C3,   A1(3).

Statistics

Order of group12
GAP identifier12,3
Presentation< k,r | k3, r2, (kr)3 >
Orders of elements1 of 1, 3 of 2, 2*4 of 3
Centre1
Derived subgroupC2×C2
Automorphism groupS4
Inner automorphism groupA4
"Out" (quotient of above)C2
Schur multiplierC2
Sylow-2-subgroupC2×C2
 

Permutation Diagrams


Sharply 2-transitive
on 4 points, even.

Sharply 4-transitive
on 4 points, even.

1-transitive on 6
points, even.

1-transitive on 6
points, even.

1-transitive on 12
points, even.

Cayley Graphs


octahemioctahedron, type I

octahemioctahedron, type I

the tetrahedron, type II


the cuboctahedron, type I

the icosahedron, type I

rectification of {6,3}(0,2), type I

Regular maps with A4 symmetry

A4 is the rotational symmetry group of the regular map the tetrahedron.