Order = 372000 = 2^{5}.3.5^{3}.31.
Mult = 1.
Out = 2.
Porting notes
Porting incomplete.Standard generators
Type I generators of L_{3}(5) are a, b where a has order 3, b is in class 5A and has order 20.
Type II generators of L_{3}(5) are x, y where x has order 2, y has order 3, xy has order 31 and xyxyy has order 5.
Standard generators of L_{3}(5):2 are c, d where c is in class 2B, d is in class 4D and cd has order 12.
Black box algorithms
Checking generators (semi-presentations)
Group | Semi-presentation | File |
---|---|---|
L_{3}(5) | 〈〈 a, b | o(a) = 3, o(b) = 5, o(ab) = 20, o(a^{2}babab) = 10 〉〉 | Download |
Presentations
Group | Presentation | Link |
---|---|---|
L_{3}(5) | 〈 a, b | a^{3} = b^{5} = aba^{−1}baba^{−1}b^{2}ab^{−2}a^{−1}b^{2} = abab^{−2}(a^{−1}b^{2}a^{−1}b^{−2})^{3} = 1 〉 | Details |
L_{3}(5) | 〈 x, y | x^{2} = y^{3} = (xy)^{31} = [x, y]^{5} = ((xy)^{5}(xy^{−1})^{4})^{2} = 1 〉 | Details |
L_{3}(5):2 | 〈 c, d | c^{2} = d^{4} = (cd)^{12} = (cdcd^{2}cd^{2})^{3} = [d^{2}, cdc]^{3} = [c, dcdcd^{−1}cdcdcd^{−1}cdcd] = 1 〉 | Details |
Representations
Representations of L_{3}(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 31 a Type I Details 31 b Type I Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 3 a Type I Details 5 GF(5) 3 b Type I Details 5 GF(5) 8 Type I Details
Representations of L_{3}(5):2
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 62 Std Details 186 Std Details 775 Std Details 3100 Std Details 3875 Std Details 4000 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 6 Std Details 5 GF(5) 8 Std Details
Maximal subgroups
Maximal subgroups of L_{3}(5)
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
5^{2}:GL_{2}(5) | 12 000 | 31 | |
5^{2}:GL_{2}(5) | 12 000 | 31 | |
S_{5} | 120 | 3 100 | |
4^{2}:S_{3} | 96 | 3 875 | |
F_{93} = 31:3 | 93 | 4 000 |
Maximal subgroups of L_{3}(5):2
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
L_{3}(5) | 372 000 | 2 | Program: Standard generators |
5^{1+2}.[2^{5}] | 4 000 | 186 | Program: Generators |
GL_{2}(5).2 | 960 | 775 | Program: Generators |
S_{5} × 2 | 240 | 3 100 | Program: Standard generators |
4^{2}:D_{12} | 192 | 3 875 | Program: Generators |
F_{186} = 31:6 | 186 | 4 000 | Program: Generators |