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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 !! Explanation
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| |-
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| | [[order of a group|order]] || [[arithmetic function value::order of a group;16|16]] ||
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| |-
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| | [[exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;8|8]] ||
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| |-
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| | [[nilpotency class]] || [[arithmetic function value::nilpotency class;2|2]] ||
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| |-
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| | [[derived length]] || [[arithmetic function value::derived length;2|2]] ||
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| |-
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| | [[Frattini length]] || [[arithmetic function value::Frattini length;3|3]] ||
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| |-
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| | [[max-length of a group|max-length]] || [[arithmetic function value::max-length of a group;4|4]] ||
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| |-
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| | [[composition length]] || [[arithmetic function value::composition length;4|4]] ||
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| | [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;2|2]] ||
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| |-
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| | [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;2|2]] ||
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| | [[rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;2|2]] ||
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| | [[normal rank of a p-group|normal rank as p-group]] || [[arithmetic function value::normal rank of a p-group;2|2]] ||
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| | [[characteristic rank of a p-group|characteristic rank as p-group]] || [[arithmetic function value::characteristic rank of a p-group;2|2]]
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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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| |-
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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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