Groups of order 8 - Revision history

R-a-jones at 23:00, 9 December 2023

2023-12-09T23:00:13Z

← Older revision		Revision as of 23:00, 9 December 2023
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	\| [[holomorph]] \|\| [[holomorph of Z8]] \|\| [[unitriangular matrix group:UT(4,2)]] \|\| [[holomorph of D8]] \|\| [[holomorph of Q8]] \|\| [[general affine group:GA(3,2)]]		\| [[holomorph]] \|\| [[holomorph of Z8]] \|\| [[unitriangular matrix group:UT(4,2)]] \|\| [[holomorph of D8]] \|\| [[holomorph of Q8]] \|\| [[general affine group:GA(3,2)]]
	\|}		\|}

			==Minimal order attaining number==

			<math>8</math> is the smallest number such that there are precisely <math>5</math> groups of that order up to isomorphism. That is, the value of the [[minimal order attaining function]] at <math>8</math> is <math>5</math>.

2016-03-05T02:02:43Z

Associated constructs

← Older revision		Revision as of 02:02, 5 March 2016
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	\| [[inner automorphism group]] \|\| [[trivial group]] \|\| [[trivial group]] \|\| [[Klein four-group]] \|\| [[Klein four-group]] \|\| [[trivial group]]		\| [[inner automorphism group]] \|\| [[trivial group]] \|\| [[trivial group]] \|\| [[Klein four-group]] \|\| [[Klein four-group]] \|\| [[trivial group]]
	\|-		\|-
	\| [[holomorph]] \|\| [[holomorph of Z8]] \|\| [[~~upper-triangular unipotent~~ matrix group:U(4,2)]] \|\| [[holomorph of D8]] \|\| [[holomorph of Q8]] \|\| [[general affine group:GA(3,2)]]		\| [[holomorph]] \|\| [[holomorph of Z8]] \|\| [[unitriangular matrix group:UT(4,2)]] \|\| [[holomorph of D8]] \|\| [[holomorph of Q8]] \|\| [[general affine group:GA(3,2)]]
	\|}		\|}

2012-08-16T22:35:02Z

Linear representation theory

← Older revision		Revision as of 22:35, 16 August 2012
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	{{further\|[[linear representation theory of groups of order 8]]}}		{{further\|[[linear representation theory of groups of order 8]]}}
			{{#lst:linear representation theory of groups of order 8\|irrep-degrees-list}}

	==Subgroup-defining functions==		==Subgroup-defining functions==

2012-08-16T22:33:25Z

Subgroup structure

← Older revision		Revision as of 22:33, 16 August 2012
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	\| [[Elementary abelian group:E8]] \|\| 5 \|\| [[subgroup structure of elementary abelian group:E8]]		\| [[Elementary abelian group:E8]] \|\| 5 \|\| [[subgroup structure of elementary abelian group:E8]]
	\|}		\|}

			==Linear representation theory==

			{{further\|[[linear representation theory of groups of order 8]]}}

	==Subgroup-defining functions==		==Subgroup-defining functions==

2012-08-16T22:21:38Z

Group properties

← Older revision		Revision as of 22:21, 16 August 2012
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	\| [[UL-equivalent group]] \|\| Yes \|\| Yes \|\| Yes \|\| Yes \|\| Yes		\| [[UL-equivalent group]] \|\| Yes \|\| Yes \|\| Yes \|\| Yes \|\| Yes
	\|-		\|-
	\| [[algebra group]] \|\| No \|\| Yes \|\| Yes \|\| ? \|\| Yes		\| [[algebra group]] \|\| No \|\| Yes \|\| Yes \|\| No \|\| Yes
	\|}		\|}

2012-08-16T19:43:56Z

Group properties

← Older revision		Revision as of 19:43, 16 August 2012
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	\|-		\|-
	\| [[UL-equivalent group]] \|\| Yes \|\| Yes \|\| Yes \|\| Yes \|\| Yes		\| [[UL-equivalent group]] \|\| Yes \|\| Yes \|\| Yes \|\| Yes \|\| Yes
			\|-
			\| [[algebra group]] \|\| No \|\| Yes \|\| Yes \|\| ? \|\| Yes
	\|}		\|}

2012-01-16T20:16:55Z

Up to Hall-Senior genus

← Older revision		Revision as of 20:16, 16 January 2012
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	\| <math>(1^3)</math> \|\| elementary abelian group \|\| [[elementary abelian group:E8]] \|\| 1 \|\| 5		\| <math>(1^3)</math> \|\| elementary abelian group \|\| [[elementary abelian group:E8]] \|\| 1 \|\| 5
	\|}		\|}

			===Up to isologism for higher class===

			Since all the groups of order 8 has class at most two, we have a unique equivalence class under isologism for any class equal to or more than two.

			===Up to isologism for elementary abelian groups===

			Each of the abelian groups is in a different equivalence class under the equivalence relation of being [[isologic groups\|isologic]] with respect to elementary abelian 2-groups. The two non-abelian groups are isologic to each other with respect to the variety of elementary abelian 2-groups.

	===Cohomology tree===		===Cohomology tree===

2012-01-16T20:13:43Z

Up to Hall-Senior genus

← Older revision		Revision as of 20:13, 16 January 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Description of ~~equivalence class~~ !! Members ~~!! Hall-Senior name~~ !! Hall-Senior numbers !! Second parts of GAP ID of members		! Genus name !! Description of genus !! Members !! Hall-Senior numbers !! Second parts of GAP ID of members
	\|-		\|-
	\| ~~Cyclic~~ group \|\| [[cyclic group:Z8]] ~~\|\| <math>(3)</math>~~ \|\| 3 \|\| 1		\| <math>(3)</math> \|\| cyclic group \|\| [[cyclic group:Z8]] \|\| 3 \|\| 1
	\|-		\|-
	\| ~~Abelian~~ group for partition <math>3 = 2 + 1</math> \|\| [[direct product of Z4 and Z2]] \|\| ~~<math>(21)</math> \|\|~~ 2 \|\| 2		\| <math>(21)</math> \|\| abelian group for partition <math>3 = 2 + 1</math> \|\| [[direct product of Z4 and Z2]] \|\| 2 \|\| 2
	\|-		\|-
	~~\| Non-abelian groups \|\| [[dihedral group:D8]], [[quaternion group]] \|~~\| <math>8\Gamma_2a</math> (the dihedral group is <math>8\Gamma_2a_1</math> and the quaternion group is <math>8\Gamma_2a_2</math>) \|\| 4,5 \|\| 3,4		\| <math>8\Gamma_2a</math> (the dihedral group is <math>8\Gamma_2a_1</math> and the quaternion group is <math>8\Gamma_2a_2</math>) \|\| non-abelian groups \|\| [[dihedral group:D8]], [[quaternion group]] \|\| 4,5 \|\| 3,4
	\|-		\|-
	\| ~~Elementary~~ abelian group \|\| [[elementary abelian group:E8]] ~~\|\| <math>(1^3)</math>~~ \|\| 1 \|\| 5		\| <math>(1^3)</math> \|\| elementary abelian group \|\| [[elementary abelian group:E8]] \|\| 1 \|\| 5
	\|}		\|}

2012-01-16T20:10:13Z

Up to isoclinism

← Older revision		Revision as of 20:10, 16 January 2012
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	\| <math>\Gamma_2</math> \|\| [[Klein four-group]] \|\| [[cyclic group:Z2]] \|\| 2 \|\| 2 \|\| [[dihedral group:D8]], [[quaternion group]] \|\| 3,4 \|\| 4,5 \|\| 8 \|\| [[dihedral group:D8]], [[quaternion group]]		\| <math>\Gamma_2</math> \|\| [[Klein four-group]] \|\| [[cyclic group:Z2]] \|\| 2 \|\| 2 \|\| [[dihedral group:D8]], [[quaternion group]] \|\| 3,4 \|\| 4,5 \|\| 8 \|\| [[dihedral group:D8]], [[quaternion group]]
	\|-		\|-
	! Total (2 rows) !! -- !! -- !! -- ~~!! 5~~ !! -- !! -- !! -- !! -- !! --		! Total (2 rows) !! -- !! -- !! 5 !! -- !!-- !! -- !! -- !! -- !! --
	\|}		\|}

2012-01-16T20:09:42Z

Up to isoclinism

← Older revision		Revision as of 20:09, 16 January 2012
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	\| <math>\Gamma_1</math> (abelian groups) \|\| [[trivial group]] \|\| [[trivial group]] \|\| 3 \|\| 1 \|\| [[cyclic group:Z8]], [[direct product of Z4 and Z2]], [[elementary abelian group:E8]] \|\| 1,2,5 \|\| 1-3 \|\| 1 \|\| [[trivial group]]		\| <math>\Gamma_1</math> (abelian groups) \|\| [[trivial group]] \|\| [[trivial group]] \|\| 3 \|\| 1 \|\| [[cyclic group:Z8]], [[direct product of Z4 and Z2]], [[elementary abelian group:E8]] \|\| 1,2,5 \|\| 1-3 \|\| 1 \|\| [[trivial group]]
	\|-		\|-
	\| <math>\Gamma_2</math> \|\| [[Klein four-group]] \|\| [[cyclic group:Z2]] \|\| 2 \|\| 2 \|\| [[dihedral group:D8]], [[quaternion group]] \|\| 3,4 \|\| 4,5 \|\| [[dihedral group:D8]], [[quaternion group]]		\| <math>\Gamma_2</math> \|\| [[Klein four-group]] \|\| [[cyclic group:Z2]] \|\| 2 \|\| 2 \|\| [[dihedral group:D8]], [[quaternion group]] \|\| 3,4 \|\| 4,5 \|\| 8 \|\| [[dihedral group:D8]], [[quaternion group]]
	\|-		\|-
	! Total (2 rows) !! -- !! -- !! -- !! 5 !! -- !! -- !! -- !! -- !! --		! Total (2 rows) !! -- !! -- !! -- !! 5 !! -- !! -- !! -- !! -- !! --