# Groups of order 1600

## Contents

See pages on algebraic structures of order 1600| See pages on groups of a particular order

## Statistics at a glance

The number 1600 has prime factors 2 and 5. The prime factorization is as follows:

$\! 1600 = 2^6 \cdot 5^2 = 64 \cdot 25$

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity Value Explanation
Number of groups up to isomorphism 10281
Number of abelian groups up to isomorphism 22 (number of abelian groups of order $2^6$) $\times$ (number of abelian groups of order $5^2$) = (number of unordered integer partitions of 6) $\times$ (number of unordered integer partitions of 2) = $11 \times 2 = 22$.
See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 534 (number of groups of order 64) $\times$ (number of groups of order 25 -- see also groups of prime-square order) = $267 \times 2 = 534$.
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 up to isomorphism  ?
Number of solvable groups up to isomorphism 10281 There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups up to isomorphism 0 Follows from all groups of this order being solvable.

## GAP implementation

The order 1600 is part of GAP's SmallGroup library. Hence, any group of order 1600 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 1600 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(1600);

There are 10281 groups of order 1600.
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 [ 50, 3 ].
1013 has Frattini factor [ 50, 4 ].
1014 has Frattini factor [ 50, 5 ].
1015 has Frattini factor [ 80, 49 ].
1016 - 1141 have Frattini factor [ 80, 50 ].
1142 - 1515 have Frattini factor [ 80, 51 ].
1516 - 1583 have Frattini factor [ 80, 52 ].
1584 has Frattini factor [ 100, 9 ].
1585 has Frattini factor [ 100, 10 ].
1586 has Frattini factor [ 100, 11 ].
1587 has Frattini factor [ 100, 12 ].
1588 - 1709 have Frattini factor [ 100, 13 ].
1710 - 1831 have Frattini factor [ 100, 14 ].
1832 - 1953 have Frattini factor [ 100, 15 ].
1954 - 2006 have Frattini factor [ 100, 16 ].
2007 - 2009 have Frattini factor [ 160, 234 ].
2010 - 2012 have Frattini factor [ 160, 235 ].
2013 - 2033 have Frattini factor [ 160, 236 ].
2034 - 2053 have Frattini factor [ 160, 237 ].
2054 - 2060 have Frattini factor [ 160, 238 ].
2061 has Frattini factor [ 200, 40 ].
2062 - 2198 have Frattini factor [ 200, 41 ].
2199 - 2295 have Frattini factor [ 200, 42 ].
2296 - 2365 have Frattini factor [ 200, 43 ].
2366 - 2382 have Frattini factor [ 200, 44 ].
2383 - 2476 have Frattini factor [ 200, 45 ].
2477 - 2570 have Frattini factor [ 200, 46 ].
2571 - 2664 have Frattini factor [ 200, 47 ].
2665 - 2756 have Frattini factor [ 200, 48 ].
2757 - 4258 have Frattini factor [ 200, 49 ].
4259 - 4860 have Frattini factor [ 200, 50 ].
4861 - 5462 have Frattini factor [ 200, 51 ].
5463 - 5599 have Frattini factor [ 200, 52 ].
5600 has Frattini factor [ 320, 1635 ].
5601 has Frattini factor [ 320, 1636 ].
5602 has Frattini factor [ 320, 1637 ].
5603 has Frattini factor [ 320, 1638 ].
5604 has Frattini factor [ 320, 1639 ].
5605 has Frattini factor [ 320, 1640 ].
5606 - 5642 have Frattini factor [ 400, 205 ].
5643 - 5657 have Frattini factor [ 400, 206 ].
5658 - 5745 have Frattini factor [ 400, 207 ].
5746 - 5772 have Frattini factor [ 400, 208 ].
5773 - 6187 have Frattini factor [ 400, 209 ].
6188 - 6489 have Frattini factor [ 400, 210 ].
6490 - 6745 have Frattini factor [ 400, 211 ].
6746 - 6785 have Frattini factor [ 400, 212 ].
6786 has Frattini factor [ 400, 213 ].
6787 - 6912 have Frattini factor [ 400, 214 ].
6913 - 7038 have Frattini factor [ 400, 215 ].
7039 - 7164 have Frattini factor [ 400, 216 ].
7165 - 7290 have Frattini factor [ 400, 217 ].
7291 - 8972 have Frattini factor [ 400, 218 ].
8973 - 9346 have Frattini factor [ 400, 219 ].
9347 - 9720 have Frattini factor [ 400, 220 ].
9721 - 9788 have Frattini factor [ 400, 221 ].
9789 - 9795 have Frattini factor [ 800, 1191 ].
9796 - 9820 have Frattini factor [ 800, 1192 ].
9821 - 9832 have Frattini factor [ 800, 1193 ].
9833 - 9868 have Frattini factor [ 800, 1194 ].
9869 - 9871 have Frattini factor [ 800, 1195 ].
9872 - 9874 have Frattini factor [ 800, 1196 ].
9875 - 9877 have Frattini factor [ 800, 1197 ].
9878 - 9890 have Frattini factor [ 800, 1198 ].
9891 - 9942 have Frattini factor [ 800, 1199 ].
9943 - 10005 have Frattini factor [ 800, 1200 ].
10006 - 10052 have Frattini factor [ 800, 1201 ].
10053 - 10064 have Frattini factor [ 800, 1202 ].
10065 - 10067 have Frattini factor [ 800, 1203 ].
10068 - 10088 have Frattini factor [ 800, 1204 ].
10089 - 10109 have Frattini factor [ 800, 1205 ].
10110 - 10130 have Frattini factor [ 800, 1206 ].
10131 - 10151 have Frattini factor [ 800, 1207 ].
10152 - 10206 have Frattini factor [ 800, 1208 ].
10207 - 10226 have Frattini factor [ 800, 1209 ].
10227 - 10246 have Frattini factor [ 800, 1210 ].
10247 - 10253 have Frattini factor [ 800, 1211 ].
10254 - 10281 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 5 of the SmallGroups library.
IdSmallGroup is available for this size.