A5

Also called icosahedral group, PSL(2,4), PSL(2,5), SL(2,4), A₁(4), A₁(5).

Statistics
Order of group	60
GAP identifier	60,5
Presentation	< p, q \| p⁵, q², (pq)² >
Orders of elements	1 of 1, 15 of 2, 20 of 3, 2*12 of 5
Centre	1
Derived subgroup	A5
Automorphism group	S5
Inner automorphism group	A5
"Out" (quotient of above)	C2
Schur multiplier	C2
Sylow-2-subgroup	C2×C2

Permutation Diagrams

Sharply 3-transitive
on 5 points, even.

Sharply 3-transitive
on 5 points, even.

Sharply 3-transitive
on 5 points, even.

2-transitive on 6
points, even.

2-transitive on 6
points, even.

1-transitive on 10
points, even.

1-transitive on 10
points, even.

1-transitive on 12
points, even.

1-transitive on 12
points, even.

1-transitive on 15
points, even.

1-transitive on 20
points, even.

1-transitive on 30
points, even.

1-transitive on 30
points, even.

Cayley Graphs

sphere/trunico

the icosahedron, type II

sphere/trundod

the dodecahedron, type II

sphere/srid

sphere/snubdod

Regular maps with A5 symmetry

A5 is the rotational symmetry group of the regular maps the dodecahedron, the icosahedron, the hemi-icosahedron, the hemidodecahedron, N5:{5,5}, S4:{5,5}, the hemi-icosidodecahedron.