Vipul: Created page with '{{particular group}} ==Definition== This group is defined by the following presentation: $G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab…'$

2009-11-03T16:31:13Z

Created page with '{{particular group}} ==Definition== This group is defined by the following presentation: <math>G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab…'

New page

{{particular group}}

==Definition==

This group is defined by the following [[presentation]]:

<math>G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle</math>

==Arithmetic functions==

{{quotation|''Want to compare with other groups of the same order? Check out [[groups of order 81#Arithmetic functions]]''}}

{| class="wikitable" border="1"
! Function !! Value !! Explanation
|-
| [[order of a group|order]] || [[arithmetic function value::order of a group;81|81]] ||
|-
| [[exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;9|9]] ||
|-
| [[Frattini length]] || [[arithmetic function value::Frattini length;2|2]] ||
|-
| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] ||
|-
| [[derived length]] || [[arithmetic function value::derived length;2|2]] ||
|-
| [[nilpotency class]] || [[arithmetic function value::nilpotency class;2|2]] ||
|-
| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;2|2]] ||
|-
| [[rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;3|3]] ||
|-
| [[normal rank of a p-group|normal rank as p-group]] || [[arithmetic function value::normal rank of a p-group;3|3]] ||
|-
| [[characteristic rank of a p-group|characteristic rank as p-group]] || [[arithmetic function value::characteristic rank of a p-group;3|3]]
|-
| [[number of conjugacy classes]] || [[arithmetic function value::number of conjugacy classes;33|33]] ||
|-
| [[number of subgroups]] || [[arithmetic function value::number of subgroups;41|41]] ||
|-
| [[number of conjugacy classes of subgroups]] || [[arithmetic function value::number of conjugacy classes of subgroups;23|23]] ||
|}

==Group properties==

{{quotation|''Want to compare with other groups of the same order? Check out [[groups of order 81#Group properties]].''}}

{| class="wikitable" border="1"
!Property !! Satisfied !! Explanation
|-
|[[satisfies property::group of prime power order]] || Yes ||
|-
|[[dissatisfies property::abelian group]] || No ||
|-
|[[satisfies property::nilpotent group]] || Yes || [[prime power order implies nilpotent]]
|-
|[[satisfies property::group of nilpotency class two]] || Yes ||
|-
|[[satisfies property::metabelian group]] || Yes ||
|-
|[[dissatisfies property::T-group]] || No ||
|-
|[[dissatisfies property::ambivalent group]] || No || [[odd-order and ambivalent implies trivial]]
|}
==Families==

This group is part of the family [[SmallGroup(p^4,3)]].

==GAP implementation==

{{GAP ID|81|3}}

===Other descriptions===

<pre>gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^9,F.2^3,F.3^3,F.1*F.2*F.1^(-1)*F.2^(-1),F.2*F.3*F.2^(-1)*F.3^(-1),F.3*F.1*F.3^(-1)*F.1^(-1)*F.2^(-1)];
<fp group on the generators [ f1, f2, f3 ]></pre>

SmallGroup(81,3) - Revision history

Vipul: Created page with '{{particular group}} ==Definition== This group is defined by the following presentation: G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab…'

Vipul: Created page with '{{particular group}} ==Definition== This group is defined by the following presentation: $G := \langle a,b,c \mid a^9 = b^3 = c^3 = e, ab = ba, bc = cb, cac^{-1} = ab…'$