ATLAS: Nonsplit extension 5^{3}.L_{3}(5)
Order = 46500000 = 2^{5}.3.5^{6}.31.
Mult = 1.
Out = 1.
The page for the image L_{3}(5) is available
here.
The following information is available for 5^{3}.L_{3}(5):
Type I standard generators for 5^{3}.L_{3}(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 L_{3}(5).
Type II standard generators for 5^{3}.L_{3}(5) are x
and 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. These generators map onto Type II standard generators of
L_{3}(5).
NB: 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).
NB: Class 5B is the unique class with centraliser order 2500.
We may obtain a conjugate of (a, b) as:
a' = y^{xyx}, b' = ((xyxyxyy)^{2}xyy)^{4}.
A presentation of 5^{3}.L_{3}(5) on its standard generators is given below.
< 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 >.
This presentation is available in Magma format as follows:
53L3(5) on x and y.
The representations of 5^{3}.L_{3}(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).

Dimension 620 over GF(2):
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).
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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.