Order = 60 = 22.3.5.
Mult = 2.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of A5 = L2(5) = L2(4) are a, b where a has order 2, b has order 3 and ab has order 5.

Standard generators of 2.A5 = SL2(5) are preimages A, B where B has order 3 and AB has order 5.

Standard generators of S5 = A5:2 are c, d where c is in class 2B, d has order 4 and cd has order 5.

Standard generators of 2.S5 are preimages C, D where CD has order 5.

Standard generators of 2.S5i are preimages C, D where CD has order 5.

## Black box algorithms

### Checking generators (semi-presentations)

Group Semi-presentation File

## Presentations

A5 a, b | a2 = b3 = (ab)5 = 1 〉 Details
2.A5 A, B | A4 = [A2, B] = B3 = (AB)5 = 1 〉 Details
S5 c, d | c2 = d4 = (cd)5 = [c, d]3 = 1 〉 Details
2.S5 C, D | C2 = D8 = [C, D4] = (CD)5 = [C, D]3 = 1 〉 Details
2.S5i C, D | C4 = C−2D4 = (CD)5 = [C, D]3 = 1 〉 Details

## Representations

### Representations of A5

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
5 Std Natural representation Details
6 Std Details
10 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
0 Z[b5] 3 a Std Details
0 Z[b5] 3 b Std Details
0 Z 4 Std Details
0 Z 5 Std Details
0 Z 6 Std Details
Char Ring Dimension ID Generators Description Link
2 GF(4) 2 a Std Natural representation as L2(4) Details
2 GF(4) 2 b Std Field automorph of the above Details
2 GF(2) 4 a Std Natural representation as O4(2) Details
2 GF(2) 4 b Std Details
Char Ring Dimension ID Generators Description Link
3 GF(9) 3 a Std Details
3 GF(9) 3 b Std Details
3 GF(3) 4 Std Details
3 GF(3) 6 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(5) 3 Std Details
5 GF(5) 5 Std Details

## Maximal subgroups

### Maximal subgroups of A5

Subgroup Order Index Programs/reps
A4 12 5 Program: Generators
D10 10 6 Program: Generators
S3 6 10 Program: Generators

### Maximal subgroups of S5

Subgroup Order Index Programs/reps
A5 60 2 Program: Standard generators
S4 24 5 Program: Generators
5:4 20 6 Program: Generators
S3 × 2 = D12 12 10 Program: Generators