Groups of order 24 - Revision history

R-a-jones: /* Similar prime structure */

2023-12-10T20:51:09Z

Similar prime structure

← Older revision		Revision as of 20:51, 10 December 2023
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	===Similar prime structure===		===Similar prime structure===

	See [[classification of groups of order eight times a prime congruent to 3 mod 4]], or more generally [[classification of groups of order eight times a prime]].		See [[classification of groups of order eight times a prime congruent to 3 mod 8]], or more generally [[classification of groups of order eight times a prime]].

	==Minimal order attaining number==		==Minimal order attaining number==

R-a-jones: /* Similar prime structure */

2023-12-10T20:50:28Z

Similar prime structure

← Older revision		Revision as of 20:50, 10 December 2023
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	===Similar prime structure===		===Similar prime structure===

	* [[~~Classification~~ of groups of order eight times a prime]]		See [[classification of groups of order eight times a prime congruent to 3 mod 4]], or more generally [[classification of groups of order eight times a prime]].

	==Minimal order attaining number==		==Minimal order attaining number==

R-a-jones: /* Relation with other orders */

2023-12-10T20:50:02Z

Relation with other orders

← Older revision		Revision as of 20:50, 10 December 2023
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	\| 10 \|\| 240 \|\| 208 \|\| [[groups of order 240]] \|\| \|\|		\| 10 \|\| 240 \|\| 208 \|\| [[groups of order 240]] \|\| \|\|
	\|}		\|}

			===Similar prime structure===

			* [[Classification of groups of order eight times a prime]]

	==Minimal order attaining number==		==Minimal order attaining number==

R-a-jones: /* The list */

2023-12-10T20:49:14Z

The list

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 There are 15 groups of order 24.
-{| class="sortable" border="1"
+{{Groups of order 24 table}}
 ==Relation with other orders==

R-a-jones at 23:19, 9 December 2023

2023-12-09T23:19:29Z

← Older revision		Revision as of 23:19, 9 December 2023
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	\| 10 \|\| 240 \|\| 208 \|\| [[groups of order 240]] \|\| \|\|		\| 10 \|\| 240 \|\| 208 \|\| [[groups of order 240]] \|\| \|\|
	\|}		\|}

			==Minimal order attaining number==

			<math>24</math> is the smallest number such that there are precisely <math>15</math> groups of that order up to isomorphism. That is, the value of the [[minimal order attaining function]] at <math>15</math> is <math>24</math>.

	==GAP implementation==		==GAP implementation==

Vipul: /* Factorization and useful forms */

2012-06-17T20:53:48Z

Factorization and useful forms

← Older revision		Revision as of 20:53, 17 June 2012
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	Other expressions for this number are:		Other expressions for this number are:

	<math>24 = 4! = 3^3 - 3 = 2^2(2^2 - 1)(2^2 - 2) = 2(2^2)(2^2 - 1)</math>		<math>24 = 4! = 3^3 - 3 = 2^2(2^2 - 1)(2^2 - 2) = 2(2^2)(2^2 - 1) = 2^3 \cdot 3</math>

	===Group counts===		===Group counts===

Vipul: /* The list */

2012-06-09T19:57:49Z

The list

← Older revision		Revision as of 19:57, 9 June 2012
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	\| [[direct product of Dic12 and Z2]] \|\| 7 \|\| not nilpotent \|\| 2		\| [[direct product of Dic12 and Z2]] \|\| 7 \|\| not nilpotent \|\| 2
	\|-		\|-
	\| [[~~SmallGroup(24,8)~~]] \|\| 8 \|\| not nilpotent \|\| 2		\| [[semidirect product of Z3 and D8 with action kernel V4]] \|\| 8 \|\| not nilpotent \|\| 2
	\|-		\|-
	\| [[direct product of Z6 and Z4]] (also, direct product of Z12 and Z2) \|\| 9 \|\| 1 \|\| 1		\| [[direct product of Z6 and Z4]] (also, direct product of Z12 and Z2) \|\| 9 \|\| 1 \|\| 1
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	\| [[direct product of E8 and Z3]] \|\| 15 \|\| 1 \|\| 1		\| [[direct product of E8 and Z3]] \|\| 15 \|\| 1 \|\| 1
	\|}		\|}


	==Relation with other orders==		==Relation with other orders==

Vipul: /* Group counts */

2012-06-07T21:16:39Z

Group counts

← Older revision		Revision as of 21:16, 7 June 2012
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	\|-		\|-
	\| Number of [[solvable group]]s (i.e., [[finite solvable group]]s) up to isomorphism \|\| {{solvable count\|15}} \|\| {{only two prime factors hence solvable}}		\| Number of [[solvable group]]s (i.e., [[finite solvable group]]s) up to isomorphism \|\| {{solvable count\|15}} \|\| {{only two prime factors hence solvable}}
			\|-
			\| Number of possible multisets of [[composition factor]]s, i.e., number of equivalence classes under [[composition factor-equivalent groups\|composition factor-equivalence]] \|\| 1 \|\| Follows from all groups of this order being solvable. The only possibility is: [[cyclic group:Z2]] (3 times), [[cyclic group:Z3]] (1 time).<br>{{composition factor multiset explanation}}
	\|-		\|-
	\| Number of simple groups \|\| 0 \|\| Follows from all groups of this order being solvable		\| Number of simple groups \|\| 0 \|\| Follows from all groups of this order being solvable

Vipul: /* Factorization and useful forms */

2012-06-07T21:14:59Z

Factorization and useful forms

← Older revision		Revision as of 21:14, 7 June 2012
Line 14:		Line 14:
	Other expressions for this number are:		Other expressions for this number are:

	<math>24 = 4! = 3^3 - 3 = 2^2(2^2 - 1)(2^2 - 2)</math>		<math>24 = 4! = 3^3 - 3 = 2^2(2^2 - 1)(2^2 - 2) = 2(2^2)(2^2 - 1)</math>

	===Group counts===		===Group counts===

Vipul: /* Group counts */

2012-05-22T17:48:11Z

Group counts

← Older revision		Revision as of 17:48, 22 May 2012
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	! Quantity !! Value !! Explanation		! Quantity !! Value !! Explanation
	\|-		\|-
	\| Total number of groups \|\| [[count::15]] \|\|		\| Total number of groups up to isomorphism \|\| [[count::15]] \|\|
	\|-		\|-
	\| Number of abelian ~~groups~~ \|\| [[abelian count::3]] \|\| (number of abelian groups of order <math>2^3</math>) times (number of abelian groups of order <math>3^1</math>) = ([[number of unordered integer partitions]] of 3) times ([[number of unordered integer partitions]] of 1) = <math>3 \times 1 = 3</math>. {{abelian count explanation}}		\| Number of [[abelian group]]s (i.e., [[finite abelian group]]s) up to isomorphism \|\| {{abelian count\|3}} \|\| (number of abelian groups of order <math>2^3</math>) times (number of abelian groups of order <math>3^1</math>) = ([[number of unordered integer partitions]] of 3) times ([[number of unordered integer partitions]] of 1) = <math>3 \times 1 = 3</math>. {{abelian count explanation}}
	\|-		\|-
	\| Number of nilpotent ~~groups~~ \|\| ~~[[abelian~~ count::5]] \|\| (number of [[groups of order 8]]) times (number of [[groups of order 3]]) = <math>5 \times 1 = 5</math>. {{nilpotent count explanation}}		\| Number of [[nilpotent group]]s (i.e., [[finite nilpotent group]]s) up to isomorphism \|\| {{nilpotent count\|5}} \|\| (number of [[groups of order 8]]) times (number of [[groups of order 3]]) = <math>5 \times 1 = 5</math>. {{nilpotent count explanation}}
	\|-		\|-
	\| Number of solvable ~~groups~~ \|\| [[solvable count::15]] \|\| {{only two prime factors hence solvable}}		\| Number of [[solvable group]]s (i.e., [[finite solvable group]]s) up to isomorphism \|\| {{solvable count\|15}} \|\| {{only two prime factors hence solvable}}
	\|-		\|-
	\| Number of simple groups \|\| 0 \|\| Follows from all groups of this order being solvable		\| Number of simple groups \|\| 0 \|\| Follows from all groups of this order being solvable