Order = 46500000 = 2^{6}.3.5^{6}.31.
Mult = 1.
Out = 1.
Porting notes
Porting complete, but handling of notes is unsatisfactory.Standard generators
Standard generators of 5^{3}.L_{3}(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 5^{3}.L_{3}(5) are x, y where x has order 2, y has order 3, xy has order 31, xyxy^{2} has order 25, (xy)^{5}(xy^{2})^{4} has order 2 and (xyxyxy^{2}xy^{2})^{2}xyxy^{2} has order 3.
Notes
- (5^{3}.L_{3}(5)) Type I standard generators map onto Type I standard generators of L_{3}(5).
- (5^{3}.L_{3}(5)) Type II standard generators map onto Type II standard generators of L_{3}(5).
- (5^{3}.L_{3}(5)) We may obtain a conjugate of (a, b) as: a' = yxyx, b' = ((xyxyxyy)^{2}xyy)^{4}.
Presentations
Group | Presentation | Link |
---|---|---|
5^{3}.L_{3}(5) | 〈 x, y | x^{2} = y^{3} = (xy)^{31} = ((xy)^{5}(xy−1)^{4})^{2} = [x, yxy(xy^{−1})^{3}xyxy(xy^{−1})^{3}xyxy] = [x, yxy]^{10} = (xy)^{6}xy^{−1}xy(xyxy^{−1})^{3}xy^{−1}xy(xy^{−1})^{5}xyxy^{−1}(xy)^{5}xy^{−1}xy^{−1} = 1 〉 | Details |
Representations
Representations of 5^{3}.L_{3}(5)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 3875 a Type II Cosets of 5^{2}:GL_{2}(5). Details 3875 b Type II Cosets of 5^{3}:4S_{4}. 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 |
---|---|---|
5^{3}.L_{3}(5) | Group theoretic trivia. | All elements of order 5 of 5^{3}.L_{3}(5) not in the normal 5^{3} map onto class 5A of L_{3}(5). |
5^{3}.L_{3}(5) | Group theoretic trivia. | Class 5B is the unique class with centraliser order 2500. |