ATLAS: Suzuki group Sz(32)

Order = 32537600 = 2¹⁰.5².31.41.
Mult = 1.
Out = 5.

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Porting notes

Porting incomplete.

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Standard generators

Standard generators of Sz(32) are a, b where a has order 2, b has order 4, ab has order 5, ab² has order 25 and abab²ab³ has order 25. Alternatively: a has order 2, b has order 4, ab has order 5, abab² has order 25 and ababab²abab²ab² has order 25.

Standard generators of Sz(32):5 are c, d where c has order 2, d has order 5, cd has order 5, cdcd⁴ has order 5, cd²cd³ has order 31 and cdcd²cd³ has order 10.

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Representations

Representations of Sz(32)

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  1025		Std		Details
  198400		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  0	Z[i, ½]	124	a	Std		Details
  0	Z[i, ½]	124	b	Std		Details
  2	GF(32)	4	a	Std		Details
  5	GF(5)	124	a	Std		Details
  41	GF(41)	124	b	Std		Details

Representations of Sz(32):5

• View detailed report.
• Permutation representations:

  Number of points	ID	Generators	Description	Link
  1025		Std		Details

• Matrix representations

  Char	Ring	Dimension	ID	Generators	Description	Link
  2	GF(2)	20		Std		Details
  5	GF(5)	124		Std		Details
  5	GF(5)	248		Std		Details
  41	GF(41)	124		Std		Details

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Maximal subgroups

Maximal subgroups of Sz(32)

Subgroup	Order	Index	Programs/reps
25+5:31			Program: Generators
41:4			Program: Generators
25:4			Program: Generators
D62			Program: Generators

Maximal subgroups of Sz(32):5

Subgroup	Order	Index	Programs/reps
Sz(32)
25+5:31:5
41:20
25:20
31:10

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Conjugacy classes

Conjugacy classes of Sz(32)

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	32 537 600		bbbb
2A	1 024		bb
4A	64		b
4B	64		(b)31
5A	25		(abb)5
25A	25		abb
25B	25		abbabbabbabbabbabbabbabb
25C	25		abbabb
25D	25		abbabbabbabbabbabbabbabbabbabbabbabbabbabbabbabb
25E	25		abbabbabbabb
31A	31		(ababbb)5
31B	31		(ababbbababbb)5
31C	31		(ababbbababbbababbbababbb)5
31D	31		(ababbbababbbababbbababbbababbbababbbababbbababbb)5
31E	31		Omitted owing to length.
31F	31		(ababbb)25
31G	31		(ababbbababbb)25
31H	31		(ababbbababbbababbbababbb)25
31I	31		(ababbbababbbababbbababbbababbbababbbababbbababbb)25
31J	31		Omitted owing to length.
31K	31		ababbb
31L	31		ababbbababbb
31M	31		ababbbababbbababbbababbb
31N	31		ababbbababbbababbbababbbababbbababbbababbbababbb
31O	31		Omitted owing to length.
41A	41		(ababbabbababbabbababbabbababbabb)3
41B	41		(ababbabbababbabbababbabbababbabb)6
41C	41		(ababbabbababbabbababbabbababbabb)12
41D	41		(ababbabbababbabbababbabbababbabb)11
41E	41		(ababbabbababbabbababbabbababbabb)7
41F	41		ababbabbababbabbababbabbababbabb
41G	41		Omitted owing to length.
41H	41		Omitted owing to length.
41I	41		ababbabb
41J	41		ababbabbababbabb

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Conjugacy classes of Sz(32):5

Conjugacy class	Centraliser order	Power up	Class rep(s)
1A	162 688 000
2A	5 120
4A	320
4B	320
5A	125
25A	25		cddcdcddcdcddcddccdcddcddc
31A	31
31B	31
31C	31
41A	41
41B	41
5B	100
5C	100
5D	100
5E	100
10A	20
10B	20
10C	20
10D	20
20A	20
20B	20
20C	20
20D	20
20E	20
20F	20
20G	20
20H	20
25B	25
25C	25
25D	25
25E	25
31A-O			cdcddcddc
41A-J			cdcddcdd
20A-B''''			cdd
25F-F''''			cdcddcdcddcddc

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Sz(32)
Sz(32):5