ATLAS: Linear group L_{3}(5)
Order = 372000 = 2^{5}.3.5^{3}.31.
Mult = 1.
Out = 2.
The page for the group 5^{3}.L_{3}(5) (nonsplit extension)
is available here.
The following information is available for L_{3}(5):
Type I standard generators of L_{3}(5) are a and b
where a has order 3, b is in class 5A and ab has
order 20.
Type II standard generators of L_{3}(5) are x and y
where x has order 2, y has order 3, xy has order 31
and xyxyy has order 5.
Standard generators of L_{3}(5):2 are c and d where
c is in class 2B, d is in class 4D and cd has order 12.
We may obtain .. as
x = ((ab)^{10})^{babb}, y = a, and
a' = a = y, b' = ((xyyxyyxyxy)^{4})^{yxyyx}.
The composition of these two maps (either way round) is equivalent to
conjugating the generators by an outer element, o say [which is the same either way round], in class 6B,
where o^{2} = a = y.
Presentations for L_{3}(5) and L_{3}(5):2 on their standard generators are given below.
< a, b  a^{3} = b^{5} = aba^{1}baba^{1}b^{2}ab^{2}a^{1}b^{2} = abab^{2}(a^{1}b^{2}a^{1}b^{2})^{3} = 1 >.
< x, y  x^{2} = y^{3} = (xy)^{31} = [x, y]^{5} = ((xy)^{5}(xy^{1})^{4})^{2} = 1 >.
< c, d  c^{2} = d^{4} = (cd)^{12} = (cdcd^{2}cd^{2})^{3} = [d^{2}, cdc]^{3} = [c, dcdcd^{1}cdcdcd^{1}cdcd] = 1 >.
These presentations are available in Magma format as follows:
L3(5) on a and b,
L3(5) on x and y and
L3(5):2 on c and d.
The representations of L_{3}(5) available are:
 Some primitive permutation representations.

Permutations on 31 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 31 points  automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some faithful irreducibles in characteristic 5.

Dimension 3 over GF(5)  the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 3 over GF(5)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of L_{3}(5):2 available are:
 Permutation representations, including all primitive ones.

Permutations on 62 points  imprimitive:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 186 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 775 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 3100 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 3875 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 4000 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 6 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 8 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of L_{3}(5) are as follows.

5^{2}:GL_{2}(5).

5^{2}:GL_{2}(5).

S_{5}.

4^{2}:S_{3}.

F_{93} = 31:3.
The maximal subgroups of L_{3}(5):2 are as follows.

L_{3}(5), with Type I standard generators
(cd)^4, ((cdcddcdcddd)^4)^(cdcdcdd).

5^{1+2}.[2^{5}], with generators
d, cddcdcdcdddc.

GL_{2}(5).2, with generators
c, ddcdcdcddd.

S_{5} × 2, with (standard) generators
cddc, d.

4^{2}:D_{12}, with generators
dd, cdcdcdddcdcdcdddcdc.

F_{186} = 31:6, with generators
c, dcddcdddcd.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L3(5) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 28th July 1999.
Last updated 04.02.02 by JNB.
Information checked to
Level 0 on 30.07.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.