Order = 46500000 = 26.3.56.31.
Mult = 1.
Out = 1.
Porting notes
Porting complete, but handling of notes is unsatisfactory.Standard generators
Standard generators of 53.L3(5) are a, b where a has order 3, b is in class 5B, ab has order 20, ababbbaabbb has order 4 and ababaababb has order 3.
Standard generators of 53.L3(5) are x, y where x has order 2, y has order 3, xy has order 31, xyxy2 has order 25, (xy)5(xy2)4 has order 2 and (xyxyxy2xy2)2xyxy2 has order 3.
Notes
- (53.L3(5)) Type I standard generators map onto Type I standard generators of L3(5).
- (53.L3(5)) Type II standard generators map onto Type II standard generators of L3(5).
- (53.L3(5)) We may obtain a conjugate of (a, b) as: a' = yxyx, b' = ((xyxyxyy)2xyy)4.
Presentations
| Group | Presentation | Link |
|---|---|---|
| 53.L3(5) | 〈 x, y | x2 = y3 = (xy)31 = ((xy)5(xy−1)4)2 = [x, yxy(xy−1)3xyxy(xy−1)3xyxy] = [x, yxy]10 = (xy)6xy−1xy(xyxy−1)3xy−1xy(xy−1)5xyxy−1(xy)5xy−1xy−1 = 1 〉 | Details |
Representations
Representations of 53.L3(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 3875 a Type II Cosets of 52:GL2(5). Details 3875 b Type II Cosets of 53:4S4. Details 4650 Type II Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 620 Type II Details 5 GF(5) 16 a Type II Uniserial 6.10. Details
Miscellaneous Notes
| Group | Category | Note |
|---|---|---|
| 53.L3(5) | Group theoretic trivia. | All elements of order 5 of 53.L3(5) not in the normal 53 map onto class 5A of L3(5). |
| 53.L3(5) | Group theoretic trivia. | Class 5B is the unique class with centraliser order 2500. |