Order = 3420 = 22.32.5.19.
Mult = 2.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of L2(19) are a, b where a has order 2, b has order 3 and ab has order 19.

Standard generators of 2.L2(19) = SL2(19) are preimages A, B where B has order 3 and AB has order 19.

Standard generators of L2(19):2 = PGL2(19) are c, d where c is in class 2B, d has order 3, cd has order 20 and cdcdd has order 5.

Standard generators of 2.L2(19).2 are preimages C, D where D has order 3.

## Black box algorithms

### Checking generators (semi-presentations)

Group Semi-presentation File

## Presentations

L2(19) a, b | a2 = b3 = (ababab−1)5 = [a, bab(ab−1)3abab] = 1 〉 Details
L2(19):2 c, d | c2 = d3 = (cd)20 = [c, d]5 = ((cd)4(cd−1)3)2 = 1 〉 Details

## Representations

### Representations of 2.L2(19)

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
40StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
19GF(19)2StdDetails

## Conjugacy classes

### Conjugacy classes of L2(19)

Conjugacy class Centraliser order Power up Class rep(s)
1A3 420 (ababb)9
2A20 (abababbabb)5
3A9 (ababb)3
5A10 abababbabbabababbabbabababbabbabababbabb
5B10 abababbabbabababbabb
9A9 ababb
9B9 ababbababb
9C9 ababbababbababbababb
10A10 abababbabb
10B10 (abababbabb)3
19A19 ab
19B19 abab

### Conjugacy classes of L2(19):2

Conjugacy class Centraliser order Power up Class rep(s)
1A6 840 (cdcdcddcdcddcdcdcddcdcdd)9
2A40 (cdcd)5
3A18 (cdcdcddcdcddcdcdcddcdcdd)3
5A20 cdcdcdcdcdcdcdcd
5B20 cdcdcdcd
9A18 cdcdcddcdcddcdcdcddcdcdd
9B18 cdcdcddcdcddcdcdcddcdcddcdcdcddcdcddcdcdcddcdcdd
9C18 Omitted owing to length.
10A20 cdcd
10B20 (cdcd)3
19A19 cdcdcddcdcdcddcdcdd
2B36 (cdcdcddcdcdd)9
4A20 (cd)5
6A18 (cdcdcddcdcdd)3
18A18 (cdcdcddcdcdd)25
18B18 (cdcdcddcdcdd)5
18C18 cdcdcddcdcdd
20A20 (cd)27
20B20 cd
20C20 (cd)3
20D20 (cd)9