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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 = 32|
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| order p-log = 5|
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| degree = 16|
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| degree p-log = 4}}
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| ==Group properties==
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| {| class="wikitable" border="1"
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| !Property !! Satisfied !! Explanation !! Comment
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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::Nilpotent group]] || Yes || [[prime power order implies nilpotent]] ||
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| |-
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| |[[Satisfies property::Metacyclic group]] || Yes || ||
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| |-
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| |[[Satisfies property::Supersolvable group]] || Yes || ||
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| |-
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| |[[Satisfies property::Solvable group]] || Yes || ||
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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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| ==Subgroup structure==
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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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| The subgroups are as follows:
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| # The trivial subgroup. Isomorphic to [[subgroup::trivial group]]. (1) | |
| # The two-element subgroup <math>\langle a^4 \rangle</math>. This is the [[derived subgroup]], and is also the [[socle]]. In particular, it is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (1)
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| # The two-element subgroups <math>\langle x \rangle</math> and <math>\langle a^4x \rangle</math>. These are [[conjugate subgroups]]. Isomorphic to [[subgroup::cyclic group:Z2]]. (2)
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| # The four-element subgroup <math>\langle a^2 \rangle</math>. This is the [[center]], and is also the [[Frattini subgroup]]. In particular, it is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
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| # The four-element subgroup <math>\langle a^2x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::cyclic group:Z4]]. (1)
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| # The four-element subgroup <math>\langle a^4, x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::Klein four-group]]. (1)
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| # The eight-element subgroups <math>\langle a \rangle</math> and <math>\langle ax \rangle</math>. These are both [[normal subgroup]]s and are [[automorphic subgroups]] -- an outer automorphism interchanges them. Isomorphic to [[subgroup::cyclic group:Z8]]. (2)
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| # The eight-element subgroup <math>\langle a^2,x \rangle</math>. This is a [[characteristic subgroup]]. Isomorphic to [[subgroup::direct product of Z4 and Z2]]. (1)
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| # The whole group. (1)
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| ==GAP implementation==
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| {{GAP ID|16|6}}
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