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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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| {| class="wikitable" border="1"
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| ! Function !! Value !! Similar groups || Explanation for function value
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| | [[underlying prime of a p-group]] || [[arithmetic function value::underlying prime of a p-group;2|2]] || ||
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| | {{arithmetic function value order|16}} ||
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| | {{arithmetic function value order p-log etc|4}}
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| | {{arithmetic function value given order and p-log|exponent of a group|8|16|4}} ||
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| | {{arithmetic function value given order and p-log|prime-base logarithm of exponent|3|16|4}} ||
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| | {{arithmetic function value given order and p-log|Frattini length|3|16|4}} ||
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| | {{arithmetic function value given order and p-log|nilpotency class|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|derived length|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|minimum size of generating set|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|subgroup rank of a group|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|rank of a p-group|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|normal rank of a p-group|2|16|4}} ||
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| | {{arithmetic function value given order and p-log|characteristic rank of a p-group|2|16|4}} ||
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| |}
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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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| |[[Dissatisfies property::Abelian group]] || No || <math>a,x</math> do not commute ||
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| |[[Satisfies property::Nilpotent group]] || Yes || [[prime power order implies nilpotent]] ||
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| |[[Satisfies property::Metacyclic group]] || Yes || ||
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| |[[Satisfies property::Supersolvable group]] || Yes || ||
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| |[[Satisfies property::Solvable group]] || Yes || ||
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| |}
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| ==GAP implementation==
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| ===Group ID===
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| This group has ID <math>6</math> among the groups of order sixteen. It can thus be defined using GAP's [[GAP:SmallGroup|SmallGroup]] function as follows:
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| <pre>SmallGroup(16,6)</pre>
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