Element structure of groups of order 64: Difference between revisions

From Groupprops
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| || 266 || [[direct product of E16 and Z4]] || 260 || [[cocycle halving generalization of Baer correspondence]], the intermediate object being a [[class two Lie cring]] || || || ||
| || 266 || [[direct product of E16 and Z4]] || 260 || [[cocycle halving generalization of Baer correspondence]], the intermediate object being a [[class two Lie cring]] || || || ||
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| [[SmallGroup(64,17)]] || 17 || [[direct product of Z8 and Z4 and Z2]] || 83 || [[cocycle skew reversal generalization of Baer correspondence]], the intermediate object being a [[class two near-Lie cring]] || || || ||
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| [[direct product of SmallGroup(32,2) and Z2]] || 56 || [[direct product of Z4 and Z4 and V4]] || 192 || [[cocycle skew reversal generalization of Baer correspondence]], the intermediate object being a [[class two near-Lie cring]] || || || ||
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| [[SmallGroup(64,113)]] || 113 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
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| [[SmallGroup(64,114)]] || 114 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
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| [[SmallGroup(64,210)]] || 210 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
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| [[semidirect product of Z16 and Z4 via fifth power map]] || 28 || [[direct product of Z16 and Z4]] || 26 || ?  || || || ||
| [[semidirect product of Z16 and Z4 via fifth power map]] || 28 || [[direct product of Z16 and Z4]] || 26 || ?  || || || ||
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| || 82 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || ||
| || 82 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || ||
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| [[SmallGroup(64,17)]] || 17 || [[direct product of Z8 and Z4 and Z2]] || 83 || [[cocycle skew reversal generalization of Baer correspondence]], the intermediate object being a [[class two near-Lie cring]] || || || ||
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| [[SmallGroup(64,25)]] || 25 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
| [[SmallGroup(64,25)]] || 25 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
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| [[SmallGroup(64,113)]] || 113 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
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| [[SmallGroup(64,114)]] || 114 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
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| [[direct product of SmallGroup(32,2) and Z2]] || 56 || [[direct product of Z4 and Z4 and V4]] || 192 || [[cocycle skew reversal generalization of Baer correspondence]], the intermediate object being a [[class two near-Lie cring]] || || || ||
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| || 61 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
| || 61 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
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| [[direct product of SmallGroup(32,33) and Z2]] || 209 || [[direct product of Z4 and Z4 and V4]] || 192 ||  || || || ||
| [[direct product of SmallGroup(32,33) and Z2]] || 209 || [[direct product of Z4 and Z4 and V4]] || 192 ||  || || || ||
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| [[SmallGroup(64,210)]] || 210 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
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Revision as of 17:26, 24 June 2011

This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 64.
View element structure of group families | View element structure of groups of a particular order |View other specific information about groups of order 64

Conjugacy class sizes

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

Grouping by conjugacy class sizes

Number of size 1 conjugacy classes Number of size 2 conjugacy classes Number of size 4 conjugacy classes Number of size 8 conjugacy classes Number of size 16 conjugacy classes Total number of conjugacy classes Total number of groups Nilpotency class(es) attained by these groups Hall-Senior family/families List of GAP IDs (second part)
64 0 0 0 0 64 11 1 , i.e., all the abelian groups of order 64 [SHOW MORE]
16 24 0 0 0 40 31 2 [SHOW MORE]
8 12 8 0 0 28 60 2,3 (class three) and (class two) [SHOW MORE]
8 0 14 0 0 22 10 2 ? [SHOW MORE]
4 30 0 0 0 34 7 2 [SHOW MORE]
4 14 0 4 0 22 23 3, 4 [SHOW MORE]
4 12 9 0 0 25 15 2 [SHOW MORE]
4 6 12 0 0 24 38 2,3 and [SHOW MORE]
4 4 9 2 0 19 31 3 [SHOW MORE]
4 2 6 4 0 16 9 3 [SHOW MORE]
4 0 15 0 0 19 5 2 [SHOW MORE]
2 15 0 0 2 19 3 5 [SHOW MORE]
2 9 11 0 0 22 3 3 [SHOW MORE]
2 5 5 4 0 16 9 3,4 [SHOW MORE]
2 3 8 3 0 16 6 3 [SHOW MORE]
2 1 5 5 0 13 6 4 [SHOW MORE]

Here is the GAP code to generate this:[SHOW MORE]

Grouping by cumulative conjugacy class sizes (number of elements)

Number of size 1 conjugacy classes Number of size 2 conjugacy classes Number of size 4 conjugacy classes Number of size 8 conjugacy classes Number of size 16 conjugacy classes Total number of conjugacy classes Total number of groups Nulpotency class(es) attained by these groups Hall-Senior family/families List of GAP IDs (second part)
64 64 64 64 64 64 11 1 , i.e., all the abelian groups of order 64 [SHOW MORE]
16 64 64 64 64 40 31 2 [SHOW MORE]
8 32 64 64 64 28 60 2,3 (class three) and (class two) [SHOW MORE]
8 8 64 64 64 22 10 2 ? [SHOW MORE]
4 64 64 64 64 34 7 2 [SHOW MORE]
4 32 32 64 64 22 23 3, 4 [SHOW MORE]
4 28 64 64 64 25 15 2 [SHOW MORE]
4 16 64 64 64 24 38 2,3 and [SHOW MORE]
4 12 48 64 64 19 31 3 [SHOW MORE]
4 8 32 64 64 16 9 3 [SHOW MORE]
4 4 64 64 64 19 5 2 [SHOW MORE]
2 32 32 32 64 19 3 5 [SHOW MORE]
2 20 64 64 64 22 3 3 [SHOW MORE]
2 12 32 64 64 16 9 3,4 [SHOW MORE]
2 8 40 64 64 16 6 3 [SHOW MORE]
2 4 24 64 64 13 6 4 [SHOW MORE]

Note that it is not true that the cumulative conjugacy class size statistics values divide the order of the group in all cases. There are a few counterexamples in the table above, as we can see values such as 12, 20, 28, and 40. is the smallest prime power where such examples exist. See also:

1-isomorphism

Pairs where one of the groups is abelian

There are 29 pairs of groups that are 1-isomorphic with the property that one of them is abelian. Of these, some pairs share the abelian group part, as the table below shows. Of these, the only example of a group that is not of nilpotency class two is SmallGroup(64,25) (GAP ID: 25):

Here is a summary version:

Nature of 1-isomorphism Intermediate object Number of 1-isomorphisms between non-abelian and abelian group of this type Number of 1-isomorphisms between non-abelian and abelian group of this nature, not of any of the preceding types Note
linear halving generalization of Baer correspondence class two Lie ring 3 3 The examples are SmallGroup(64,57), direct product of SmallGroup(32,4) and Z2 (ID: (64,84)), and semidirect product of Z16 and Z4 of M-type (ID: (64,27)). See table below.
cocycle halving generalization of Baer correspondence class two Lie cring 17 14 See table below.
cocycle skew reversal generalization of Baer correspondence class two near-Lie cring 19 2 The two new examples are SmallGroup(64,17) and direct product of SmallGroup(32,2) and Z2.
linear halving generalization of Lazard correspondence class three Lie ring 4 0 Of the four examples, three arise via the linear halving generalization of Baer correspondence. The fourth, namely semidirect product of Z8 and Z8 of M-type (ID: (64,3)), alternatively arises via the cocycle halving generalization of Baer correspondence.
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] ? ?

Here is a long version:

Non-abelian member of pair GAP ID Abelian member of pair GAP ID The 1-isomorphism arises as a ... Description of the 1-isomorphism Best perspective 1 Best perspective 2 Alternative perspective
semidirect product of Z16 and Z4 of M-type 27 direct product of Z16 and Z4 26 linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
SmallGroup(64,57) 57 direct product of Z4 and Z4 and Z4 55 linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
direct product of SmallGroup(32,4) and Z2 84 direct product of Z8 and Z4 and Z2 83 linear halving generalization of Baer correspondence, the intermediate object being a class two Lie ring second cohomology group for trivial group action of direct product of Z4 and Z4 on Z4
semidirect product of Z8 and Z8 of M-type 3 direct product of Z8 and Z8 2 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring; alternatively, the linear halving generalization of Lazard correspondence, the intermediate object being a class three Lie ring
M64 51 direct product of Z32 and Z2 50 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M16 and Z4 85 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
central product of M16 and Z8 over common Z2 86 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
112 direct product of Z8 and Z4 and Z2 83 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M32 and Z2 184 direct product of Z16 and V4 183 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
central product of D8 and Z16 185 direct product of Z16 and V4 183 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(32,24) and Z2 195 direct product of Z4 and Z4 and V4 192 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(16,13) and Z4 198 direct product of Z4 and Z4 and V4 192 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of M16 and V4 247 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
SmallGroup(64,248) 248 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
249 direct product of Z8 and E8 246 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
direct product of SmallGroup(16,13) and V4 263 direct product of E16 and Z4 260 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
266 direct product of E16 and Z4 260 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
SmallGroup(64,17) 17 direct product of Z8 and Z4 and Z2 83 cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring
direct product of SmallGroup(32,2) and Z2 56 direct product of Z4 and Z4 and V4 192 cocycle skew reversal generalization of Baer correspondence, the intermediate object being a class two near-Lie cring
SmallGroup(64,113) 113 direct product of Z8 and Z4 and Z2 83 ?
SmallGroup(64,114) 114 direct product of Z8 and Z4 and Z2 83 ?
SmallGroup(64,210) 210 direct product of Z4 and Z4 and V4 192 ?
semidirect product of Z16 and Z4 via fifth power map 28 direct product of Z16 and Z4 26 ?
64 direct product of Z4 and Z4 and Z4 55 ?
82 direct product of Z4 and Z4 and Z4 55 ?
SmallGroup(64,25) 25 direct product of Z8 and Z4 and Z2 83 ?
61 direct product of Z4 and Z4 and V4 192 ?
77 direct product of Z4 and Z4 and V4 192 ?
direct product of SmallGroup(32,33) and Z2 209 direct product of Z4 and Z4 and V4 192
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