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
5StdNatural representationDetails
6StdDetails
10StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
0Z[b5]3aStdDetails
0Z[b5]3bStdDetails
0Z4StdDetails
0Z5StdDetails
0Z6StdDetails
Char Ring Dimension ID Generators Description Link
2GF(4)2aStdNatural representation as L2(4)Details
2GF(4)2bStdField automorph of the aboveDetails
2GF(2)4aStdNatural representation as O4(2)Details
2GF(2)4bStdDetails
Char Ring Dimension ID Generators Description Link
3GF(9)3aStdDetails
3GF(9)3bStdDetails
3GF(3)4StdDetails
3GF(3)6StdDetails
Char Ring Dimension ID Generators Description Link
5GF(5)3StdDetails
5GF(5)5StdDetails

## 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