ATLAS: Linear group L3(5)
Order = 372000 = 25.3.53.31.
Mult = 1.
Out = 2.
The page for the group 53.L3(5) (non-split extension)
is available here.
The following information is available for L3(5):
Type I standard generators of L3(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 L3(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 L3(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 L3(5) and L3(5):2 on their standard generators are given below.
< a, b | a3 = b5 = aba-1baba-1b2ab-2a-1b2 = abab-2(a-1b2a-1b-2)3 = 1 >.
< x, y | x2 = y3 = (xy)31 = [x, y]5 = ((xy)5(xy-1)4)2 = 1 >.
< c, d | c2 = d4 = (cd)12 = (cdcd2cd2)3 = [d2, cdc]3 = [c, dcdcd-1cdcdcd-1cdcd] = 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 L3(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 L3(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 L3(5) are as follows.
-
52:GL2(5).
-
52:GL2(5).
-
S5.
-
42:S3.
-
F93 = 31:3.
The maximal subgroups of L3(5):2 are as follows.
-
L3(5), with Type I standard generators
(cd)^4, ((cdcddcdcddd)^4)^(cdcdcdd).
-
51+2.[25], with generators
d, cddcdcdcdddc.
-
GL2(5).2, with generators
c, ddcdcdcddd.
-
S5 × 2, with (standard) generators
cddc, d.
-
42:D12, with generators
dd, cdcdcdddcdcdcdddcdc.
-
F186 = 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.