Order = 960 = 2^{6}.3.5.

Mult = 4^{2} × 2.

Out = S_{4}.

## Porting notes

Porting incomplete.## Standard generators

Standard generators of M_{20} =
2^{4}:A_{5} are *a*, *b* where
*a* has order 4, *b* has order 3 and *a**b*
has order 4.

Standard generators of 2_{1}.M_{20} =
2^{1+4}:A_{5} are preimages *A*, *B*
where *B* has order 3, *A**B* has order 5,
*A**B**B* has order 10 and
*A**A**B* has order 6.

Standard generators of 2_{2}.M_{20} =
2^{4}:SL_{2}(5) are preimages *A*,
*B* where *B* has order 3 and *A**B* has
order 5.

Standard generators of 2_{3}.M_{20} are
*A*, *B* where *B* has order 3, *A**B*
has order 5, *A**B**B* has order 5 and
*A**A**B* has order 3.

Standard generators of 4_{1}.M_{20} are
preimages *A*, *B* where *B* has order 3,
*A**B* has order 5, *A* has order 4 and
*A**B**A**B**A**A**A**B**A**B**B*
has order 2.

Standard generators of 4_{2}.M_{20} are
preimages *A*, *B* where *B* has order 3,
*A**B* has order 5, *A* has order 4 and
*A**B**A**B**A**A**A**B**A**B**B*
has order 2.

### Notes

- (M
_{20}) Standard generators of M_{20}map to standard generators of A_{5}. - (4
_{1}.M_{20}) We haven't fully checked that the standard generators for 4.M20 are sufficient to define it up to automorphisms. - (4
_{2}.M_{20}) We haven't fully checked that the standard generators for 4.M20 and are sufficient to define it up to automorphisms.

## Presentations

Group | Presentation | Link |
---|---|---|

M_{20} |
〈 a, b |
a^{4} = b^{3} =
(ab)^{5} =
(ab^{−1})^{5} =
(a^{2}b)^{3} =
(abab^{−1}ab^{−1}a^{−1}b^{−1})^{2}
= 1 〉 |
Details |

## Representations

### Representations of
M_{20}

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 20 a Std Details 20 b Std Details 20 c Std Details 20 d Std Details

### Representations of
2_{1}.M_{20}

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 12 a Std Details 12 b Std Details 20 a Std Details 20 b Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 6 a Std Details 0 Z 6 b Std Details 0 Z 10 a Std Details 0 Z 10 b Std Details 0 Z[b _{5}]12 a Std Details 0 Z[b _{5}]12 b Std Basis 0. Details 0 Z[b _{5}]12 b Std Basis 1. Details 0 Z 20 Std Details 0 Z 24 Std Details

### Representations of
2_{2}.M_{20}

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 120 a Std Details

### Representations of
2_{3}.M_{20}

- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 24 Std Details

### Representations of
4_{2}.M_{20}

- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[i] 4 a Std Basis 0. Details 0 Z[i] 4 a Std Basis 1. Details 0 Z[i] 20 a Std Details