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| ==Definition==
| | #redirect [[Modular maximal-cyclic group:M16]] |
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| The group, sometimes denoted <math>M_{16}</math>, is defined as follows:
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| <math>M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle</math>.
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| Here, <math>e</math> denotes the identity element.
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| ==Arithmetic functions==
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| {{M-type 2-group arithmetic function table|
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| order = 16|
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| order p-log = 4|
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| degree = 8|
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| degree p-log = 3}}
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| ==Group properties==
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| {| class="sortable" border="1"
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| !Property !! Satisfied? !! Explanation !! Comment
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| |-
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| | {{group properties because p-group}}
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| |-
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| |[[Dissatisfies property::Abelian group]] || No || <math>a,x</math> do not commute ||
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| |-
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| |[[Satisfies property::Metacyclic group]] || Yes || ||
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| |-
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| |[[Satisfies property::Finite group that is 1-isomorphic to an abelian group]] || Yes || via [[cocycle halving generalization of Baer correspondence]] || See [[element structure of groups of order 16#1-isomorphism]]
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| |}
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| ==Elements==
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| {{further|[[element structure of M16]]}}
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| ===1-isomorphism===
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| The group is [[1-isomorphic groups|1-isomorphic]] to the group [[direct product of Z8 and Z2]]. In other words, there is a bijection between the groups that restricts to an isomorphism on all cyclic subgroups on either side. The 1-isomorphism is explained by the [[cocycle halving generalization of Baer correspondence]], where the intermediary is a [[class two Lie cring]].
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| ==Subgroups==
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| {{further|[[subgroup structure of M16]]}}
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| To describe subgroups, we use the defining presentation given at the beginning:
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| <math>M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle</math>.
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| {{#lst:subgroup structure of M16|summary}}
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| ==GAP implementation==
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| {{GAP ID|16|6}}
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