Order = 4680000 =
26.32.54.13.
Mult = 2.
Out = 2.
Porting notes
Fully copied from version 2. 29/7/06.Standard generators
Standard generators of S4(5) are a, b where a is in class 2B, b is in class 3B and ab is in class 13C. Alternatively: a is in class 2B, b is in class 3B, ab has order 13 and ababb has order 12.
Standard generators of 2.S4(5) are preimages A, B where B has order 3 and AB has order 13.
Standard generators of S4(5):2 are c, d where c is in class 2D, d is in class 3B and cd is in class 26A. Alternatively: c is in class 2D, d is in class 3B, cd has order 26 and cdcdd has order 10.
Standard generators of 2.S4(5):2 are preimages C, D where D has order 3.
Standard generators of 2.S4(5).2 (isoclinic) are preimages C, D where D has order 3.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| S4(5) | 〈〈 a, b | some conditions 〉〉 | Download |
Representations
Representations of S4(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 156 a Std Details 156 b Std Details 300 Std Details 325 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 13 a Std Details 0 C 13 b Std Details 0 Z 40 Std Details 0 Z 65 a Std Details 0 Z 65 b Std Details 0 C 78 a Std Details 0 C 78 b Std Details 0 Z 90 Std Details 0 Z 104 a Std Details 0 Z 104 b Std Details 0 Z 104 c Std Details 0 Z 130 Std Details 0 Z 156 Std Details 0 Z 416 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 12 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 13 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 5 Std Details
Representations of 2.S4(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 624 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 4 Std Details
Representations of S4(5):2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 24 Std Details 2 GF(2) 40 Std Details 2 GF(2) 64 Std Details 2 GF(2) 104 a Std Details 2 GF(2) 104 b Std Details 3 GF(3) 26 Std Details 5 GF(5) 5 Std Details
Maximal subgroups
Maximal subgroups of S4(5)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 51+2:4A5 | Program: Generators |
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| 53:(2 × A5).2 | Program: Generators |
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| L2(25):22 | Program: Generators |
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| 2.(A5 × A5).2 | Program: Generators |
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| 24:A5 | Program: Generators |
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| S3 × S5 | Program: Generators |
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| (2 × 2 × A5):2 | Program: Generators |
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| A6 | Program: Generators |