Dihedral group:D12 - Revision history

R-a-jones at 16:28, 12 January 2024

2024-01-12T16:28:10Z

2023-12-29T04:17:29Z

Group properties

← Older revision		Revision as of 04:17, 29 December 2023
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	\|[[Satisfies property::complete group]] \|\| Yes \|\| \|\|		\|[[Satisfies property::complete group]] \|\| Yes \|\| \|\|
	\|-		\|-
	\|[[Satisfies property::Group isomorphic to its automorphism group]] \|\| Yes \|\| \|\|		\|[[Satisfies property::Group isomorphic to its automorphism group]] \|\| Yes \|\| \|\| Being a [[complete group]] is a stronger property
	\|}		\|}

2023-12-29T04:17:03Z

@@ Line 40: / Line 40: @@
 |-
 | [[max-length of a group|max-length]] || [[arithmetic function value::max-length of a group;3|3]]
 |}

2012-09-12T23:33:27Z

← Older revision		Revision as of 23:33, 12 September 2012
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	* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].		* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].
	* It is the [[outer linear group]] of degree two over the field of two elements, i.e., the group <math>OL(2,2)</math>.		* It is the [[outer linear group]] of degree two over the field of two elements, i.e., the group <math>OL(2,2)</math>.
			* It is [[Borel subgroup of general linear group]] for [[general linear group:GL(2,3)]], i.e., the [[general linear group of degree two]] over [[field:F3]].

	The usual presentation is:		The usual presentation is:

2011-06-21T00:27:36Z

Arithmetic functions

← Older revision		Revision as of 00:27, 21 June 2011
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	==Arithmetic functions==		==Arithmetic functions==

	{\| class="~~wikitable~~" border="1"		{{compare and contrast arithmetic functions\|order = 12}}

			{\| class="sortable" border="1"
	! Function !! Value !! Explanation		! Function !! Value !! Explanation
	\|-		\|-

2011-05-23T21:47:59Z

GAP implentation

2011-05-23T21:47:45Z

GAP implentation

← Older revision		Revision as of 21:47, 23 May 2011
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	==GAP implentation==		==GAP implentation==

	~~===Group~~ ID~~===~~		{{GAP ID\|12\|4}}

	~~The group has ID <math>4</math> among the groups of order <math>~~12~~</math>. It can be created using GAP's [[GAP:SmallGroup~~\|~~SmallGroup]] function:~~

	~~<pre>SmallGroup(12,~~4~~)</pre>~~

	===Other definitions===		===Other definitions===

	~~The group can also be defined using GAP's~~ [[GAP::DihedralGroup\|DihedralGroup]] ~~command:~~		{\| class="sortable" border="1"
			! Description !! Functions used
	<~~pre~~>~~DihedralGroup~~(12)</~~pre~~>		\|-
			\| <tt>DihedralGroup(12)</tt> \|\| [[GAP:DihedralGroup\|DihedralGroup]]
			\|-
			\| <tt>DirectProduct(SymmetricGroup(3),CyclicGroup(2))</tt> \|\| [[GAP:DirectProduct\|DirectProduct]], [[GAP:SymmetricGroup\|SymmetricGroup]], [[GAP:CyclicGroup\|CyclicGroup]]
			\|}

2009-08-26T21:36:36Z

← Older revision		Revision as of 21:36, 26 August 2009
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	\| [[max-length of a group\|max-length]] \|\| [[arithmetic function value::max-length of a group;3\|3]]		\| [[max-length of a group\|max-length]] \|\| [[arithmetic function value::max-length of a group;3\|3]]
	\|}		\|}

			==GAP implentation==

			===Group ID===

			The group has ID <math>4</math> among the groups of order <math>12</math>. It can be created using GAP's [[GAP:SmallGroup\|SmallGroup]] function:

			<pre>SmallGroup(12,4)</pre>

			===Other definitions===

			The group can also be defined using GAP's [[GAP::DihedralGroup\|DihedralGroup]] command:

			<pre>DihedralGroup(12)</pre>

2009-08-26T21:32:56Z

Definition

← Older revision		Revision as of 21:32, 26 August 2009
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	* It is the [[member of family::dihedral group]] of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.		* It is the [[member of family::dihedral group]] of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.
	* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].		* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].
			* It is the [[outer linear group]] of degree two over the field of two elements, i.e., the group <math>OL(2,2)</math>.

	The usual presentation is:		The usual presentation is:

2009-08-20T20:31:11Z

← Older revision		Revision as of 20:31, 20 August 2009
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	* It is the [[member of family::dihedral group]] of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.		* It is the [[member of family::dihedral group]] of order twelve. In other words, it is the dihedral group of degree six, i.e., the group of symmetries of a regular hexagon.
	* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].		* It is the [[direct product]] of [[symmetric group:S3\|the symmetric group of degree three]] and [[cyclic group:Z2\|the cyclic group of order two]].

			The usual presentation is:

			<math>\langle a,x \mid a^6 = x^2 = e, xax = a^{-1} \rangle</math>.

			With this presentation, the symmetric group of degree three is the direct factor <math>\langle a^2,x \rangle</math> and the complement of order two is the subgroup <math>\langle a^3 \rangle</math>.

	==Arithmetic functions==		==Arithmetic functions==