ATLAS: Linear group L₃(8)

Order = 16482816 = 2⁹.3².7².73.
Mult = 1.
Out = 6.

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Porting notes

Porting incomplete.

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Standard generators

Standard generators of L₃(8) are a, b where a has order 2, b has order 3 and ab has order 21.

Standard generators of L₃(8).2 are c, d where c is in class 2B, d has order 3, cd has order 8, cdcdd has order 63 and cdcdcdd has order 8.

Standard generators of L₃(8).3 are e, f where e has order 2, f has order 12, ef has order 21, efeff has order 7, efefeff has order 12, efefefeff has order 12 and efefeffefeffeffeff has order 9.

Standard generators of L₃(8).6 are g, h where g is in class 2A, h is in class 18D, gh has order 24, ghghh has order 14, ghghghh has order 3 and ghghghghghhghghhghh has order 6.

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Black box algorithms

Checking generators (semi-presentations)

Group	Semi-presentation	File
L3(8)	〈〈 a, b | some conditions 〉〉	Download

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Representations

Representations of L₃(8)

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  73	a	Std		Details
  73	b	Std		Details
  56064		Std		Details
  75264		Std		Details
  98112		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  0	Z	72		Std		Details
  0	C	73	a	Std		Details
  0	C	73	b	Std		Details
  0	C	73	c	Std		Details
  0	C	73	d	Std		Details
  0	C	73	e	Std		Details
  0	C	73	f	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(8)	3		Std		Details
  2	GF(8)	8		Std		Details
  2	GF(8)	9	a	Std		Details
  2	GF(8)	9	b	Std		Details
  2	GF(8)	24	a	Std		Details
  2	GF(8)	24	b	Std		Details
  2	GF(2)	27		Std		Details
  2	GF(8)	27		Std		Details
  2	GF(8)	64		Std		Details
  2	GF(8)	72	a	Std		Details
  2	GF(8)	72	b	Std		Details
  2	GF(8)	192		Std		Details
  2	GF(2)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(3)	72		Std		Details
  3	GF(3)	511		Std		Details
  3	GF(27)	657		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  7	GF(7)	72		Std		Details
  7	GF(7)	511		Std		Details
  7	GF(343)	511		Std		Details
  7	GF(7)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  73	GF(73)	71		Std		Details
  73	GF(73)	441		Std		Details
  73	GF(73)	511	a	Std		Details
  73	GF(73)	511	b	Std		Details

Representations of L₃(8).2

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  146		Std		Details
  657		Std		Details
  4672		Std		Details
  56064		Std		Details
  75264		Std		Details
  98112		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(8)	6		Std		Details
  2	GF(8)	8		Std		Details
  2	GF(8)	18	a	Std		Details
  2	GF(8)	18	b	Std		Details
  2	GF(8)	48	a	Std		Details
  2	GF(8)	48	b	Std		Details
  2	GF(2)	54		Std		Details
  2	GF(8)	54		Std		Details
  2	GF(8)	64		Std		Details
  2	GF(8)	144	a	Std		Details
  2	GF(8)	144	b	Std		Details
  2	GF(8)	384		Std		Details
  2	GF(2)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(9)	72		Std		Details
  3	GF(3)	511		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  7	GF(7)	72		Std		Details
  7	GF(7)	511		Std		Details
  7	GF(7)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  73	GF(73)	71		Std		Details
  73	GF(73)	441		Std		Details
  73	GF(73)	511	a	Std		Details
  73	GF(73)	511	b	Std		Details

Representations of L₃(8).3

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  73	a	Std		Details
  73	b	Std		Details
  219	a	Std		Details
  219	b	Std		Details
  56064		Std		Details
  75264		Std		Details
  98112		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	9		Std		Details
  2	GF(2)	24		Std		Details
  2	GF(2)	27	a	Std		Details
  2	GF(2)	27	b	Std		Details
  2	GF(2)	27	c	Std		Details
  2	GF(2)	72	a	Std		Details
  2	GF(2)	72	b	Std		Details
  2	GF(2)	81		Std		Details
  2	GF(2)	192		Std		Details
  2	GF(2)	216	a	Std		Details
  2	GF(2)	216	b	Std		Details
  2	GF(2)	512		Std		Details
  2	GF(2)	576		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(3)	72		Std		Details
  3	GF(3)	511		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  7	GF(7)	72		Std		Details
  7	GF(7)	511		Std		Details
  7	GF(7)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  73	GF(73)	71		Std		Details
  73	GF(73)	441		Std		Details
  73	GF(73)	511	a	Std		Details

Representations of L₃(8).6

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  438		Std		Details
  657		Std		Details
  4672		Std		Details
  56064		Std		Details
  75264		Std		Details
  98112		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	18		Std		Details
  2	GF(2)	24		Std		Details
  2	GF(2)	54	a	Std		Details
  2	GF(2)	54	b	Std		Details
  2	GF(2)	54	c	Std		Details
  2	GF(2)	144	a	Std		Details
  2	GF(2)	144	b	Std		Details
  2	GF(2)	162		Std		Details
  2	GF(2)	192		Std		Details
  2	GF(2)	432	a	Std		Details
  2	GF(2)	432	b	Std		Details
  2	GF(2)	512		Std		Details
  2	GF(2)	1152		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(9)	72		Std		Details
  3	GF(3)	511		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  7	GF(7)	72		Std		Details
  7	GF(7)	511		Std		Details
  7	GF(7)	512		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  73	GF(73)	71		Std		Details
  73	GF(73)	441		Std		Details
  73	GF(73)	511	a	Std		Details

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Maximal subgroups

Maximal subgroups of L₃(8)

Subgroup	Order	Index	Programs/reps
26:(7xL2(8))			Program: Generators
26:(7xL2(8))			Program: Generators
72:S3			Program: Generators
73:3			Program: Generators
L2(7)			Program: Generators

Maximal subgroups of L₃(8).2

Subgroup	Order	Index	Programs/reps
L3(8)			Program: Generators
2(3+6):72:2			Program: Generators
D14xL2(8)			Program: Generators
72:D12			Program: Generators
73:6			Program: Generators
L2(7):2			Program: Generators

Maximal subgroups of L₃(8).3

Subgroup	Order	Index	Programs/reps
L3(8)			Program: Generators
26:(7xL2(8)):3			Program: Generators
26:(7xL2(8)):3			Program: Generators
72:(3xS3)			Program: Generators
73:9			Program: Generators
L2(7)x3			Program: Generators

Maximal subgroups of L₃(8).6

Subgroup	Order	Index	Programs/reps
L3(8).3			Program: Generators
L3(8).2			Program: Generators
2(3+6):72:6			Program: Generators
(D14xL2(8)):3			Program: Generators
72:(6xS3)			Program: Generators
73:18			Program: Generators
L2(7):2x3			Program: Generators

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Conjugacy classes

Conjugacy classes of L₃(8)

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	16 482 816
2A	3 584
3A	63
4A	64		abbabababbababbabbabababbabbabbabbabababbababb
7A	3 528
7B	3 528
7C	3 528
7D	3 528
7E	3 528
7F	3 528
7G	49		abbabababbabbabbabbabababbababb
7H	49
7I	49		(ababb)4
7J	49
7K	49
9A	63
9B	63
9C	63
14A	56		(abbabbabbabababbababb)9
14B	56
14C	56
14D	56
14E	56
14F	56
21A	63
21B	63
21C	63
21D	63
21E	63
21F	63
63A	63		(ababababbabb)43
63B	63
63C	63
63D	63
63E	63
63F	63
63G	63
63H	63
63I	63
63J	63
63K	63
63L	63
63M	63
63N	63
63O	63
63P	63
63Q	63
63R	63
73A	73		(abababbab)12
73B	73
73C	73
73D	73
73E	73
73F	73
73G	73
73H	73
73I	73
73J	73
73K	73
73L	73
73M	73
73N	73
73O	73
73P	73
73Q	73
73R	73
73S	73
73T	73
73U	73
73V	73
73W	73
73X	73

Download words for class representatives.

Conjugacy classes of L₃(8).2

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	32 965 632
2A	7 168
3A	126
4A	128
7A	3 528
7B	3 528
7C	3 528
7D	49
7E	98
7F	98
7G	98		cdcdcdcddcdcdcddcddcdcdd
9A	126
9B	126
9C	126
14A	56		(cdcdcddcddcdcdcddcdcddcdd)3
14B	56
14C	56
21A	63
21B	63
21C	63
63A	63		(cdcdd)10
63B	63
63C	63
63D	63
63E	63
63F	63
63G	63
63H	63
63I	63
73A	73		(cdcdcddcdcddcdd)7
73B	73
73C	73
73D	73
73E	73
73F	73
73G	73
73H	73
73I	73
73J	73
73K	73
73L	73
2B	1 008
6A	18
8A	32
8B	32
14D	14
14E	14
14F	14
18A	18
18B	18
18C	18
8A-B			cd
14G-I			cdcdcddcdcddcddcdd
18A-C			cdcdcdcddcdd

Download words for class representatives.

Conjugacy classes of L₃(8).3

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	49 448 448
2A	10 752
3A	189
4A	192
7A	3 528
7D	3 528
7G	147
7H	147
7I	49		efefefeffeffefefefeffeffefefeff
9A	63
14A	56		efefeffeffefefefefeffeffeffeff
14B	56
21A	63
21D	63
63A	63		(efefefefeffeffeffeff)23
63B	63
63C	63
63D	63
63E	63
63F	63
73A	73		(efefefeffefeffeffeff)9
73B	73
73C	73
73D	73
73E	73
73F	73
73G	73
73H	73
3B	504
3C	504
6A	24
6B	24
9D	9		efefefefeffeff
9D'	9
12A	12		f
12B	12
21G	21		ef
21G'	21
21H	21
21H'	21

Download words for class representatives.

Conjugacy classes of L₃(8).6

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	98 896 896
2A	21 504
3A	378
4A	384
7A	3 528
7B	147
7C	98
9A	126
14A	56		ghghghghhghhghhghghh
21A	63
63A	63		(ghghghghhghghhghhghh)5
63B	63
63C	63
73A	73		(ghghghhghh)3
73B	73
73C	73
73D	73
2B	3 024
6A	54
8A	96
8B	96
14G	14		ghghh
18A	18		ghghghghhghhghh
3B	1 008
3C	1 008
6B	48
6C	48		ghghghghghhghghhghh
9B	18
9C	18
12A	24
12B	24
21G	21		ghghhghghghhghhghhghghh
21C	21
6D	36
6E	36
18D	18		h
18D'	18
24A	24
24A'	24
24B	24
24B'	24
24A-B			gh

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L₃(8)
L₃(8).2
L₃(8).3
L₃(8).6