S4

Also called  A1(3),   full tetrahedral group,   octahedral group,   GL(2,3).

Statistics

Order of group24
GAP identifier24,12
Presentation< k,r | k4, r2, (kr)3 >
Orders of elements1 of 1, 3+6 of 2, 8 of 3, 6 of 4
Centre1
Derived subgroupA4
Automorphism groupS4
Inner automorphism groupS4
"Out" (quotient of above)1
Schur multiplierC2
Sylow-2-subgroupD8
 

Permutation Diagrams


Sharply 4-transitive
on 4 points, odd.

Sharply 4-transitive
on 4 points, odd.

Sharply 4-transitive
on 4 points, odd.

Sharply 4-transitive
on 4 points, odd.

1-transitive on 6
points, even.

1-transitive on 6
points, even.

1-transitive on 6
points, even.

1-transitive on 6
points, even.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 6
points, odd.

1-transitive on 8
points, even.

1-transitive on 12
points, even.

1-transitive on 12
points, odd.

Cayley Graphs




the tetrahedron, type III

the cube, type II

the octahedron, type II


{6,3}(0,4), type I

{6,3}(2,2), type II

Regular maps with S4 symmetry

S4 is the rotational symmetry group of the regular maps the octahedron,   the cube,   the hemioctahedron,   the hemicube,   the cuboctahedron,   the hemi-cuboctahedron,   small cubicuboctahedron.

S4 is the full symmetry group of the regular map the tetrahedron.


Index to regular maps