ATLAS: Linear group L_{2}(19)
Order = 3420 = 2^{2}.3^{2}.5.19.
Mult = 2.
Out = 2.
The following information is available for L_{2}(19):
Standard generators of L_{2}(19) are a and b where
a has order 2, b has order 3 and ab has order 19.
Standard generators of the double cover 2.L_{2}(19) = SL_{2}(19)
are preimages A and B where B has order 3
and AB has order 19.
Standard generators of L_{2}(19):2 = PGL_{2}(19) are c
and d where c is in class 2B, d has order 3, cd has
order 20 and cdcdd has order 5.
Standard generators of either of the double covers 2.L_{2}(19).2 =
2.PGL_{2}(19) are preimages C and D where D has
order 3.
An outer automorphism of L_{2}(19) of order 2 may be obtained by
mapping (a, b) to (a, b^{1}).
To find standard generators of L_{2}(19):
 Find any element of order 2, x say, by taking a suitable power of any element of even order.
[The probability of success at each attempt is 1 in 4.]
 Find any element of order 3, y say, by taking a suitable power of any element of order divisible by 3.
[The probability of success at each attempt is 4 in 9 (about 1 in 2).]
 Find a conjugate a of x and a conjugate b of y such that ab has order 19.
[The probability of success at each attempt is 2 in 19 (about 1 in 10).]
 Now a and b are standard generators of L_{2}(19).
To find standard generators of L_{2}(19).2:
 Find any element of order 6 or 18. This powers up to x in class 2B.
[The probability of success at each attempt is 2 in 9 (about 1 in 5) OR 4 in 9 (about 1 in 2) if we restrict our search to outer elements only.]
 Find any element of order 3, y say, by taking a suitable power of any element of order divisible by 3.
[The probability of success at each attempt is 4 in 9 (about 1 in 2).]
 Find a conjugate c of x and a conjugate d of y such that cd has order 20 and cdcdd has order 5.
[The probability of success at each attempt is 9 in 95 (about 1 in 11).]
 Now c and d are standard generators of L_{2}(19):2.
Presentations of L_{2}(19) and L_{2}(19):2 = PGL_{2}(19)
on their standard generators are given below.
< a, b  a^{2} = b^{3} = (ababab^{1})^{5} = [a, bab(ab^{1})^{3}abab] = 1 >.
< c, d  c^{2} = d^{3} = (cd)^{20} = [c, d]^{5} = ((cd)^{4}(cd^{1})^{3})^{2} = 1 >.
The representations of L_{2}(19) available are:
 All primitive permutation representations.

Permutations on 20 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 57 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 57 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 171 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 190 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles in characteristic 2:

Dimension 9 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles in characteristic 3:

Dimension 9 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 19 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles in characteristic 5:

Dimension 9 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles over GF(19):

Dimension 3 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 5 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 7 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 11 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 17 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 19 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.L_{2}(19) = SL_{2}(19) available are:

Permutations on 40 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 2 over GF(19):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of L_{2}(19):2 = PGL_{2}(19) available are:
 Permutation representations, including all faithful primitive ones.

Permutations on 20 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 114 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 171 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 190 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 285 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 3 over GF(19):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.L_{2}(19):2 (NB not the Atlas group) available are:

Dimension 20 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 20 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 2 over GF(19):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of L2(19) are as follows.
 19:9, with generators
???.
 A_{5}
 A_{5}
 D_{20}
 D_{18}
The maximal subgroups of L2(19):2 are as follows.
 L_{2}(19)
 19:18
 D_{40}
 D_{36}
 S_{4}
A set of generators for the maximal cyclic subgroups of L2(19) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L2(19):2 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L2(19) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 file created on 18th January 2002,
from Version 1 file last modified on 22.12.98.
Last updated 28.01.02 by RAW.
Information checked to
Level 0 on 18.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.