ATLAS: Linear group L₂(23)

Order = 6072 = 2³.3.11.23.
Mult = 2.
Out = 2.

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Porting notes

Porting incomplete.

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Standard generators

Standard generators of L₂(23) are a, b where a has order 2, b has order 3 and ab has order 23.

Standard generators of 2.L₂(23) = SL₂(23) are preimages A, B where B has order 3 and AB has order 23.

Standard generators of L₂(23):2 are c, d where c has order 2, d has order 3, cd has order 22 and cdcdd has order 4.

Standard generators of 2.L₂(23).2 are C, D where D has order 3.

Standard generators of 2.L₂(23).2 (isoclinic) are preimages C, D where D has order 3.

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Black box algorithms

Checking generators (semi-presentations)

Group	Semi-presentation	File
L2(23)	〈〈 a, b | some conditions 〉〉	Download

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Representations

Representations of L₂(23)

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  24		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  0	C	11	a	Std		Details
  0	C	11	b	Std		Details
  0	Z	22	a	Std		Details
  0	Z	22	b	Std		Details
  0	Z	22	c	Std		Details
  0	C	22	d	Std		Details
  0	C	22	e	Std		Details
  0	Z	23		Std		Details
  0	C	24	a	Std		Details
  0	C	24	b	Std		Details
  0	C	24	c	Std		Details
  0	C	24	d	Std		Details
  0	C	24	e	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	11	a	Std		Details
  2	GF(2)	11	b	Std		Details
  2	GF(2)	22		Std		Details
  2	GF(32)	24	a	Std		Details
  2	GF(32)	24	b	Std		Details
  2	GF(32)	24	c	Std		Details
  2	GF(32)	24	d	Std		Details
  2	GF(32)	24	e	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(3)	22	a	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  11	GF(11)	23		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  23	GF(23)	3		Std		Details
  23	GF(23)	5		Std		Details
  23	GF(23)	7		Std		Details
  23	GF(23)	9		Std		Details
  23	GF(23)	11		Std		Details
  23	GF(23)	13		Std		Details
  23	GF(23)	15		Std		Details
  23	GF(23)	17		Std		Details
  23	GF(23)	19		Std		Details
  23	GF(23)	21		Std		Details
  23	GF(23)	23		Std		Details

Representations of 2.L₂(23)

• View detailed report.
• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  23	GF(23)	2		Std		Details

Representations of L₂(23):2

• View detailed report.
• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	22	a	Std		Details
  23	GF(23)	3	a	Std		Details

Representations of 2.L₂(23).2

• View detailed report.
• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  23	GF(23)	2		Std		Details

Representations of 2.L₂(23).2 (isoclinic)

• View detailed report.
• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	23		Std		Details

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Maximal subgroups

Maximal subgroups of L₂(23)

Subgroup	Order	Index	Programs/reps
23:11
S4
S4
D24
D22

Maximal subgroups of L₂(23):2

Subgroup	Order	Index	Programs/reps
L2(23)
23:22
D48
D44

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Conjugacy classes

Conjugacy classes of L₂(23)

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	6 072		(ababbababbababbababb)3
2A	24		(ababbababb)3
3A	12		ababbababbababbababb
4A	12		(ababb)3
6A	12		ababbababb
11A	11		abababbabababb
11B	11		Omitted owing to length.
11C	11		abababbabababbabababbabababb
11D	11		abababb
11E	11		abababbabababbabababbabababbabababbabababbabababbabababb
12A	12		(ababb)5
12B	12		ababb
23A	23		ab
23B	23		(ab)11

Download words for class representatives.

Conjugacy classes of L₂(23):2

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	12 144		(cdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcdd)3
2A	48		(cdcdcddcdcdcddcdcdcddcdcdcdd)3
3A	24		cdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcddcdcdcdd
4A	24		(cdcdcddcdcdcdd)3
6A	24		cdcdcddcdcdcddcdcdcddcdcdcdd
11A	22		cdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd
11B	22		cdcdcdcd
11C	22		Omitted owing to length.
11D	22		cdcdcdcdcdcdcdcd
11E	22		cdcd
12A	24		cdcdcddcdcdcdd
12B	24		(cdcdcddcdcdcdd)5
23A	23		ccdcdcdcddcdcddcddcdcdcddcd
2B	44		(cd)11
8A	24		(cdcdcdd)3
8B	24		(cdcdcdd)9
22A	22		(cd)27
22B	22		(cd)81
22C	22		cd
22D	22		(cd)3
22E	22		(cd)9
24A	24		(cdcdcdd)77
24B	24		(cdcdcdd)11
24C	24		(cdcdcdd)55
24D	24		cdcdcdd

Download words for class representatives.

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Standard generators
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L₂(23)
2.L₂(23)
L₂(23):2
2.L₂(23).2
2.L₂(23).2 (isoclinic)