Groups of order 320 - Revision history

Vipul: /* GAP implementation */

2011-06-15T15:20:58Z

GAP implementation

← Older revision		Revision as of 15:20, 15 June 2011
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	==GAP implementation==		==GAP implementation==

	~~The~~ order ~~320 is part of GAP's [[~~GAP~~:SmallGroup~~\|~~SmallGroup]] library. Hence, all groups of~~ order 320 ~~can be constructed using the [[GAP:SmallGroup~~\|~~SmallGroup]] function and have group IDs. Also, [[GAP:IdGroup\|IdGroup]] is available, so the group ID of any group of this order can be queried.~~		{{this order on GAP\|order = 320\|idgroup = yes}}

	~~Here is GAP's summary information about how it stores groups of this order:~~

	<pre>gap> SmallGroupsInformation(320);		<pre>gap> SmallGroupsInformation(320);

Vipul at 14:49, 15 June 2011

2011-06-15T14:49:32Z

← Older revision		Revision as of 14:49, 15 June 2011
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	\| number of [[nilpotent group]]s up to isomorphism \|\| 267 \|\| (number of [[groups of order 64]]) times (number of [[groups of order 5]]) = <math>267 \times 1 = 267</math>		\| number of [[nilpotent group]]s up to isomorphism \|\| 267 \|\| (number of [[groups of order 64]]) times (number of [[groups of order 5]]) = <math>267 \times 1 = 267</math>
	\|-		\|-
	\| number of [[solvable group]]s up to isomorphism \|\| 1640 \|\| There are only two prime factors, and [[order has only two prime factors implies solvable]], so all groups of order 80 are [[solvable group]]s (specifically, [[finite solvable group]]s).		\| number of [[solvable group]]s up to isomorphism \|\| 1640 \|\| There are only two prime factors, and [[order has only two prime factors implies solvable]], so all groups of order 320 are [[solvable group]]s (specifically, [[finite solvable group]]s).
	\|-		\|-
	\| number of [[simple group]]s up to isomorphism \|\| 0 \|\| All groups of this order are solvable, so there cannot be any simple groups.		\| number of [[simple group]]s up to isomorphism \|\| 0 \|\| All groups of this order are solvable, so there cannot be any simple groups.

Vipul: Created page with "{{groups of order|320}} ==Statistics at a glance== The number 320 has prime factorization $320 = 2^6 \cdot 5$ . There are only two prime factors, and [[order has only..."

2011-06-15T14:41:10Z

Created page with "{{groups of order|320}} ==Statistics at a glance== The number 320 has prime factorization <math>320 = 2^6 \cdot 5</math>. There are only two prime factors, and [[order has only..."

New page

{{groups of order|320}}

==Statistics at a glance==

The number 320 has prime factorization <math>320 = 2^6 \cdot 5</math>. There are only two prime factors, and [[order has only two prime factors implies solvable]], so all groups of order 320 are [[solvable group]]s (specifically, [[finite solvable group]]s).

{| class="sortable" border="1"
! Quantity !! Value !! Explanation
|-
| number of groups up to isomorphism || 1640 ||
|-
| number of [[abelian group]]s up to isomorphism || 11 || (number of abelian groups of order <math>2^6</math>) times (number of abelian groups of order <math>5^1</math>) = ([[number of unordered integer partitions]] of 6) times ([[number of unordered integer partitions]] of 1) = <math>11 \times 1 = 11</math>. See also [[classification of finite abelian groups]]
|-
| number of [[nilpotent group]]s up to isomorphism || 267 || (number of [[groups of order 64]]) times (number of [[groups of order 5]]) = <math>267 \times 1 = 267</math>
|-
| number of [[solvable group]]s up to isomorphism || 1640 || There are only two prime factors, and [[order has only two prime factors implies solvable]], so all groups of order 80 are [[solvable group]]s (specifically, [[finite solvable group]]s).
|-
| number of [[simple group]]s up to isomorphism || 0 || All groups of this order are solvable, so there cannot be any simple groups.
|}

==GAP implementation==

The order 320 is part of GAP's [[GAP:SmallGroup|SmallGroup]] library. Hence, all groups of order 320 can be constructed using the [[GAP:SmallGroup|SmallGroup]] function and have group IDs. Also, [[GAP:IdGroup|IdGroup]] is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

<pre>gap> SmallGroupsInformation(320);

There are 1640 groups of order 320.
They are sorted by their Frattini factors.
1 has Frattini factor [ 10, 1 ].
2 has Frattini factor [ 10, 2 ].
3 has Frattini factor [ 20, 3 ].
4 - 125 have Frattini factor [ 20, 4 ].
126 - 178 have Frattini factor [ 20, 5 ].
179 - 272 have Frattini factor [ 40, 12 ].
273 - 874 have Frattini factor [ 40, 13 ].
875 - 1011 have Frattini factor [ 40, 14 ].
1012 has Frattini factor [ 80, 49 ].
1013 - 1138 have Frattini factor [ 80, 50 ].
1139 - 1512 have Frattini factor [ 80, 51 ].
1513 - 1580 have Frattini factor [ 80, 52 ].
1581 - 1583 have Frattini factor [ 160, 234 ].
1584 - 1586 have Frattini factor [ 160, 235 ].
1587 - 1607 have Frattini factor [ 160, 236 ].
1608 - 1627 have Frattini factor [ 160, 237 ].
1628 - 1634 have Frattini factor [ 160, 238 ].
1635 - 1640 have trivial Frattini subgroup.

For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
LGLength, FrattinifactorSize and FrattinifactorId.

This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.</pre>

Groups of order 320 - Revision history

Vipul: /* GAP implementation */

Vipul at 14:49, 15 June 2011

Vipul: Created page with "{{groups of order|320}} ==Statistics at a glance== The number 320 has prime factorization 320 = 2^6 \cdot 5. There are only two prime factors, and [[order has only..."

Vipul: Created page with "{{groups of order|320}} ==Statistics at a glance== The number 320 has prime factorization $320 = 2^6 \cdot 5$ . There are only two prime factors, and [[order has only..."