Element structure of groups of order 64: Difference between revisions

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order = 64}}
order = 64}}


==1-isomorphism==
===Pairs where one of the groups is abelian===
===Pairs where one of the groups is abelian===


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{| class="sortable" border="1"
{| class="sortable" border="1"
! Non-abelian member of pair !! GAP ID !! Abelian member of pair !! GAP ID !! Type of the 1-isomorphism !! Long explanation for the 1-isomorphism !! Description of the 1-isomorphism !! Best perspective 1 !! Best perspective 2 !! Alternative perspective  
! 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 Z8 and Z8 of M-type]] || 3 || [[direct product of Z8 and Z8]] || 2 || via class two Lie cring || || || || ||
| [[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]]|| || || ||
|-
|-
| [[semidirect product of Z16 and Z4 of M-type]] || 27 || [[direct product of Z16 and Z4]] || 26 || via class two Lie cring||  || || || ||
| [[semidirect product of Z16 and Z4 of M-type]] || 27 || [[direct product of Z16 and Z4]] || 26 || [[cocycle halving generalization of Baer correspondence]], the intermediate object being a [[class two Lie cring]] || || || ||
|-
|-
| [[M64]] || 51 || [[direct product of Z32 and Z2]] || 50 || via class two Lie cring || || || || ||
| [[M64]] || 51 || [[direct product of Z32 and Z2]] || 50 || [[cocycle halving generalization of Baer correspondence]], the intermediate object being a [[class two Lie cring]] || || || ||
|-
|-
| [[SmallGroup(64,57)]] || 57 || [[direct product of Z4 and Z4 and Z4]] || 55 || via class two Lie cring || || || || ||
| [[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]] || || || ||
|-
|-
| [[direct product of SmallGroup(32,4) and Z2]] || 84 || [[direct product of Z8 and Z4 and Z2]] || 83 || via class two Lie cring || || || || ||
| [[direct product of SmallGroup(32,4) and Z2]] || 84 || [[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 M16 and Z4]] || 85 || [[direct product of Z8 and Z4 and Z2]] || 83 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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 || via 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]] || || || ||
|-
|-
| [[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 || ? || || || ||
|-
|-
| || 64 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || || ||
| || 64 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || ||
|-
|-
| || 82 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || || ||
| || 82 || [[direct product of Z4 and Z4 and Z4]] || 55 || ? || || || ||
|-
|-
| || 17 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || || ||
| || 17 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
|-
|-
| || 25 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || || ||
| || 25 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
|-
|-
| || 113 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || || ||
| || 113 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
|-
|-
| || 114 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || || ||
| || 114 || [[direct product of Z8 and Z4 and Z2]] || 83 || ? || || || ||
|-
|-
| [[direct product of SmallGroup(32,2) and Z2]] || 56 || [[direct product of Z4 and Z4 and V4]] || 192 || via 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]] || || || ||
|-
|-
| || 61 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || || ||
| || 61 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
|-
|-
| || 77 || [[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 || via class three Lie cring || || || || ||
| [[direct product of SmallGroup(32,33) and Z2]] || 209 || [[direct product of Z4 and Z4 and V4]] || 192 || via class three Lie cring(?) || || || ||
|-
|-
| || 210 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || || ||
| || 210 || [[direct product of Z4 and Z4 and V4]] || 192 || ? || || || ||
|}
|}
|}

Revision as of 18:41, 5 April 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

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):

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 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
semidirect product of Z16 and Z4 of M-type 27 direct product of Z16 and Z4 26 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
M64 51 direct product of Z32 and Z2 50 cocycle halving generalization of Baer correspondence, the intermediate object being a class two Lie cring
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
direct product of SmallGroup(32,4) and Z2 84 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 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
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 ?
17 direct product of Z8 and Z4 and Z2 83 ?
25 direct product of Z8 and Z4 and Z2 83 ?
113 direct product of Z8 and Z4 and Z2 83 ?
114 direct product of Z8 and Z4 and Z2 83 ?
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
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 via class three Lie cring(?)
210 direct product of Z4 and Z4 and V4 192 ?
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