# Groups of order 960

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## Contents

This article gives information about, and links to more details on, groups of order 960
See pages on algebraic structures of order 960| See pages on groups of a particular order

## Statistics at a glance

### Factorization and useful forms

The number 960 has prime factors 2, 3, and 5, and prime factorization: $\! 960 = 2^6 \cdot 3^1 \cdot 5^1 =64\cdot 3 \cdot 5$

Other useful expressions for this number are: $\! 960 = 2(5^2 - 1)(5^2 - 5) = 2^3 \cdot 5!$

### Group counts

Quantity Value Explanation
Total number of groups up to isomorphism 11394
Total number of abelian groups (i.e., finite abelian groups) up to isomorphism 11 (Number of abelian groups of order $2^6$) times (Number of abelian groups of order $3^1$) times (Number of abelian groups of order $5^1$) = (number of unordered integer partitions of 6) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $11 \times 1 \times 1 = 11$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Total number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 267 (Number of groups of order 64) times (Number of groups of order 3) times (Number of groups of order 5) = $267 \times 1 \times 1 = 267$. See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
Number of supersolvable groups (i.e., finite supersolvable groups) up to isomorphism 10723
Total number of solvable groups (i.e., finite solvable groups) up to isomorphism 11287 See note on non-solvable groups
Number of non-solvable groups up to isomorphism 107 All the non-solvable groups have alternating group:A5 as the unique simple non-abelian composition factor and four cyclic group:Z2s as the other composition factors
Number of simple groups up to isomorphism 0
Number of quasisimple groups up to isomorphism 0
Number of almost simple groups up to isomorphism 0
Number of almost quasisimple groups up to isomorphism 0
Number of semisimple groups up to isomorphism 0
Number of perfect groups up to isomorphism 2 One of the groups is inner automorphism group of wreath product of Z2 and A5. This is the smallest order for which there are examples to show that perfect not implies semisimple.

## GAP implementation

The order 960 is part of GAP's SmallGroup library. Hence, any group of order 960 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 960 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(960);

There are 11394 groups of order 960.
They are sorted by their Frattini factors.
1 has Frattini factor [ 30, 1 ].
2 has Frattini factor [ 30, 2 ].
3 has Frattini factor [ 30, 3 ].
4 has Frattini factor [ 30, 4 ].
5 has Frattini factor [ 60, 6 ].
6 has Frattini factor [ 60, 7 ].
7 - 215 have Frattini factor [ 60, 8 ].
216 - 217 have Frattini factor [ 60, 9 ].
218 - 339 have Frattini factor [ 60, 10 ].
340 - 461 have Frattini factor [ 60, 11 ].
462 - 583 have Frattini factor [ 60, 12 ].
584 - 636 have Frattini factor [ 60, 13 ].
637 - 639 have Frattini factor [ 120, 34 ].
640 - 642 have Frattini factor [ 120, 35 ].
643 - 779 have Frattini factor [ 120, 36 ].
780 - 787 have Frattini factor [ 120, 37 ].
788 - 795 have Frattini factor [ 120, 38 ].
796 - 812 have Frattini factor [ 120, 39 ].
813 - 906 have Frattini factor [ 120, 40 ].
907 - 1000 have Frattini factor [ 120, 41 ].
1001 - 3720 have Frattini factor [ 120, 42 ].
3721 - 3737 have Frattini factor [ 120, 43 ].
3738 - 4339 have Frattini factor [ 120, 44 ].
4340 - 4941 have Frattini factor [ 120, 45 ].
4942 - 5543 have Frattini factor [ 120, 46 ].
5544 - 5680 have Frattini factor [ 120, 47 ].
5681 - 5725 have Frattini factor [ 240, 189 ].
5726 - 5747 have Frattini factor [ 240, 190 ].
5748 has Frattini factor [ 240, 191 ].
5749 - 5754 have Frattini factor [ 240, 192 ].
5755 - 5761 have Frattini factor [ 240, 193 ].
5762 - 5835 have Frattini factor [ 240, 194 ].
5836 - 6250 have Frattini factor [ 240, 195 ].
6251 - 6298 have Frattini factor [ 240, 196 ].
6299 - 6346 have Frattini factor [ 240, 197 ].
6347 - 6401 have Frattini factor [ 240, 198 ].
6402 has Frattini factor [ 240, 199 ].
6403 - 6528 have Frattini factor [ 240, 200 ].
6529 - 6654 have Frattini factor [ 240, 201 ].
6655 - 9638 have Frattini factor [ 240, 202 ].
9639 - 9666 have Frattini factor [ 240, 203 ].
9667 - 9672 have Frattini factor [ 240, 204 ].
9673 - 10046 have Frattini factor [ 240, 205 ].
10047 - 10420 have Frattini factor [ 240, 206 ].
10421 - 10794 have Frattini factor [ 240, 207 ].
10795 - 10862 have Frattini factor [ 240, 208 ].
10863 - 10883 have Frattini factor [ 480, 1186 ].
10884 - 10892 have Frattini factor [ 480, 1187 ].
10893 has Frattini factor [ 480, 1188 ].
10894 - 10908 have Frattini factor [ 480, 1189 ].
10909 has Frattini factor [ 480, 1190 ].
10910 - 10924 have Frattini factor [ 480, 1191 ].
10925 - 10939 have Frattini factor [ 480, 1192 ].
10940 - 10994 have Frattini factor [ 480, 1193 ].
10995 - 10997 have Frattini factor [ 480, 1194 ].
10998 - 11000 have Frattini factor [ 480, 1195 ].
11001 - 11003 have Frattini factor [ 480, 1196 ].
11004 - 11066 have Frattini factor [ 480, 1197 ].
11067 - 11087 have Frattini factor [ 480, 1198 ].
11088 - 11108 have Frattini factor [ 480, 1199 ].
11109 - 11115 have Frattini factor [ 480, 1200 ].
11116 - 11122 have Frattini factor [ 480, 1201 ].
11123 - 11143 have Frattini factor [ 480, 1202 ].
11144 - 11148 have Frattini factor [ 480, 1203 ].
11149 - 11151 have Frattini factor [ 480, 1204 ].
11152 - 11172 have Frattini factor [ 480, 1205 ].
11173 - 11193 have Frattini factor [ 480, 1206 ].
11194 - 11273 have Frattini factor [ 480, 1207 ].
11274 - 11282 have Frattini factor [ 480, 1208 ].
11283 - 11287 have Frattini factor [ 480, 1209 ].
11288 - 11307 have Frattini factor [ 480, 1210 ].
11308 - 11327 have Frattini factor [ 480, 1211 ].
11328 - 11347 have Frattini factor [ 480, 1212 ].
11348 - 11354 have Frattini factor [ 480, 1213 ].
11355 - 11394 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.