# ATLAS: Non­split extension 53.L3(5)

Order = 46500000 = 25.3.56.31.
Mult = 1.
Out = 1.
The page for the image L3(5) is available here.

The following information is available for 53.L3(5):

### Standard generators

Type I standard generators for 53.L3(5) are a and b where a has order 3, b is in class 5B, ab has order 20, abb has order 31, ababbbaabbb has order 4 and ababaababb has order 3. These generators map onto Type I standard generators of L3(5).
Type II standard generators for 53.L3(5) are x and 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. These generators map onto Type II standard generators of L3(5).

NB: All elements of order 5 of 53.L3(5) not in the normal 53 map onto class 5A of L3(5).
NB: Class 5B is the unique class with centraliser order 2500.

We may obtain a conjugate of (a, b) as: a' = yxyx, b' = ((xyxyxyy)2xyy)4.

### Presentation

A presentation of 53.L3(5) on its standard generators is given below.

< 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 >.

This presentation is available in Magma format as follows: 53L3(5) on x and y.

### Representations

The representations of 53.L3(5) available are:
• Faithful permutation representations (minimal degree 3875; 2 such).
• Permutations on 3875a points - the cosets of 5^2:GL2(5): x and y (Meataxe), x and y (Meataxe binary), x and y (GAP).
• Permutations on 3875b points - the cosets of 5^3:4S4: x and y (Meataxe), x and y (Meataxe binary), x and y (GAP).
• Permutations on 4650 points: x and y (Meataxe), x and y (Meataxe binary), x and y (GAP).
• Faithful representations in characteristic 5 (minimal degree 16; 2 such).
• Dimension 16 over GF(5) - uniserial 6.10: x and y (Meataxe), x and y (Meataxe binary), x and y (GAP).
• Faithful representations in characteristic other than 5 (minimal degree 620; unique such in each characteristic).

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Version 2.0 created on 13th July 1999.
Last updated 08.02.02 by JNB.
Information checked to Level 0 on 20.07.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.