S4

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

Statistics
Order of group	24
GAP identifier	24,12
Presentation	< k,r \| k⁴, r², (kr)³ >
Orders of elements	1 of 1, 3+6 of 2, 8 of 3, 6 of 4
Centre	1
Derived subgroup	A4
Automorphism group	S4
Inner automorphism group	S4
"Out" (quotient of above)	1
Schur multiplier	C2
Sylow-2-subgroup	D8

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

sphere/srco

sphere/snubcube

sphere/trunoct2

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.

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

Index to regular maps