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, *a**b* has order 20,
*a**b**a**b**b**b**a**a**b**b**b*
has order 4 and
*a**b**a**b**a**a**b**a**b**b*
has order 3.

Standard generators of 5^{3}.L_{3}(5) are
*x*, *y* where *x* has order 2, *y* has
order 3, *x**y* has order 31,
*x**y**x**y*^{2} has order 25,
(*x**y*)^{5}(*x**y*^{2})^{4}
has order 2 and
(*x**y**x**y**x**y*^{2}*x**y*^{2})^{2}*x**y**x**y*^{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. |