Element structure of groups of order 64
This article gives specific information, namely, element structure, about a family of groups, namely: groups of order 64.
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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:
| 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 |
|---|---|---|---|---|---|---|---|---|---|
| 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 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 via fifth power map | 28 | direct product of Z16 and Z4 | 26 | ? | |||||
| M64 | 51 | direct product of Z32 and Z2 | 50 | via class two Lie cring | |||||
| SmallGroup(64,57) | 57 | direct product of Z4 and Z4 and Z4 | 55 | via class two Lie cring | |||||
| 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 M16 and Z4 | 85 | 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 | via class two Lie cring |