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

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
3875aType IICosets of 52:GL2(5).Details
3875bType IICosets of 53:4S4.Details
4650Type IIDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(2)620Type IIDetails
5GF(5)16aType IIUniserial 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.