ATLAS: Linear group L₂(32)

Order = 32736 = 2⁵.3.11.31.
Mult = 1.
Out = 5.

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Porting notes

Porting incomplete.

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Standard generators

Standard generators of L₂(32) are a, b where a has order 2, b has order 3 and ab is in class 31A/B/C/D/E. Alternatively: a has order 2, b has order 3, ab has order 31 and ababb has order 11.

Standard generators of L₂(32):5 are c, d where c has order 2, d has order 5, cd has order 15, cdd has order 15, cdcdd has order 15, cdcdcdd has order 15 and cdcdcdcddcdcddcdd has order 11.

Notes

• (L₂(32):5) The conditions ensure that d is conjugate to the field automorphism x -> x¹⁶.

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Black box algorithms

Checking generators (semi-presentations)

Group	Semi-presentation	File
L2(32)	〈〈 a, b | some conditions 〉〉	Download

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Presentations

Group	Presentation	Link
L2(32)	〈 a, b | a2 = b3 = (ab)31 = (ab)4(ab−1abab−1)3(ab)4(ab−1)2 = 1 〉	Details
L2(32):5	〈 c, d | c2 = d5 = cdcdcd−2cd−1cd−2cdcd−2cd−1 cd2cd−2 = (cd)4cd−2(cd2)3cd−2cd2cd−1cdcd2 = 1 〉	Details

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Representations

Representations of L₂(32)

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  33		Std		Details
  496		Std		Details
  528		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  0	Z	31	a	Std		Details
  0	Z	32		Std		Details
  0	Z[z31]	33	a	Std		Details
  0	C	33	b	Std		Details
  0	C	33	c	Std		Details
  0	C	33	d	Std		Details
  0	C	33	e	Std		Details
  0	C	33	f	Std		Details
  0	C	33	g	Std		Details
  0	C	33	h	Std		Details
  0	C	33	i	Std		Details
  0	C	33	j	Std		Details
  0	C	33	k	Std		Details
  0	C	33	l	Std		Details
  0	C	33	m	Std		Details
  0	C	33	n	Std		Details
  0	C	33	o	Std		Details
  0	Z	155		Std		Details
  0	Z	310		Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(32)	2	a	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  3	GF(3)	31	a	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  11	GF(11)	31	b	Std		Details
  Char	Ring	Dimension	ID	Generators	Description	Link
  31	GF(31)	32		Std		Details

Representations of L₂(32):5

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  33		Std		Details
  496		Std		Details
  528		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	10		Std		Details
  2	GF(2)	20	a	Std		Details
  2	GF(2)	20	b	Std		Details
  2	GF(2)	32		Std		Details
  2	GF(2)	40	a	Std		Details
  2	GF(2)	40	b	Std		Details
  2	GF(2)	80		Std		Details

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Conjugacy classes

Conjugacy classes of L₂(32)

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	32 736		aa
2A	32		a
3A	33		(ababababb)11
11A	33		(ababababbababababbababababbababababb)3
11B	33		Omitted owing to length.
11C	33		Omitted owing to length.
11D	33		(ababababb)3
11E	33		(ababababbababababb)3
31A	31		ab
31B	31		abab
31C	31		abababab
31D	31		abababababababab
31E	31		abababababababababababababababab
31F	31		(abababab)9
31G	31		(abababababababab)9
31H	31		(abababababababababababababababab)9
31I	31		(ab)9
31J	31		(abab)9
31K	31		(abab)3
31L	31		(abababab)3
31M	31		(abababababababab)3
31N	31		(abababababababababababababababab)3
31O	31		(ab)3
33A	33		(ababababb)10
33B	33		(ababababb)13
33C	33		(ababababb)7
33D	33		(ababababb)14
33E	33		(ababababb)5
33F	33		ababababb
33G	33		ababababbababababb
33H	33		ababababbababababbababababbababababb
33I	33		Omitted owing to length.
33J	33		Omitted owing to length.

Download words for class representatives.

Conjugacy classes of L₂(32):5

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	163 680		(cdcddcddd)10
2A	160		(cdcddcddd)5
3A	165		(cd)5
11A	33		(cdcdcddd)3
31A	31		(cdcddcdd)3
31F	31		cdcddcdd
31K	31		(cdcddcdd)9
33A	33		cdcdcddd
33F	33		(cdcdcddd)5
5A	30		(cdcddcddd)16
5A'	30		(cdcddcddd)2
5A''	30		(cdcddcddd)8
5A'''	30		(cdcddcddd)4
10A	10		cdcddcddd
10A'	10		(cdcddcddd)27
10A''	10		(cdcddcddd)3
10A'''	10		(cdcddcddd)9
15A	15		cd
15A'	15		(cd)2
15A''	15		(cd)8
15A'''	15		(cd)4

Download words for class representatives.

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L₂(32)
L₂(32):5