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
L2(19) 〈〈 a, b | some conditions 〉〉 Download

Presentations

Group Presentation Link
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
40 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
19 GF(19) 2 Std Details

Conjugacy classes

Conjugacy classes of L2(19)

Conjugacy class Centraliser order Power up Class rep(s)
1A 3 420 (ababb)9
2A 20 (abababbabb)5
3A 9 (ababb)3
5A 10 abababbabbabababbabbabababbabbabababbabb
5B 10 abababbabbabababbabb
9A 9 ababb
9B 9 ababbababb
9C 9 ababbababbababbababb
10A 10 abababbabb
10B 10 (abababbabb)3
19A 19 ab
19B 19 abab

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Conjugacy classes of L2(19):2

Conjugacy class Centraliser order Power up Class rep(s)
1A 6 840 (cdcdcddcdcddcdcdcddcdcdd)9
2A 40 (cdcd)5
3A 18 (cdcdcddcdcddcdcdcddcdcdd)3
5A 20 cdcdcdcdcdcdcdcd
5B 20 cdcdcdcd
9A 18 cdcdcddcdcddcdcdcddcdcdd
9B 18 cdcdcddcdcddcdcdcddcdcddcdcdcddcdcddcdcdcddcdcdd
9C 18 Omitted owing to length.
10A 20 cdcd
10B 20 (cdcd)3
19A 19 cdcdcddcdcdcddcdcdd
2B 36 (cdcdcddcdcdd)9
4A 20 (cd)5
6A 18 (cdcdcddcdcdd)3
18A 18 (cdcdcddcdcdd)25
18B 18 (cdcdcddcdcdd)5
18C 18 cdcdcddcdcdd
20A 20 (cd)27
20B 20 cd
20C 20 (cd)3
20D 20 (cd)9

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