# Groups of order 1100

## Contents

See pages on algebraic structures of order 1100| See pages on groups of a particular order

## Statistics at a glance

The number 1100 has prime factors 2, 5, and 11. The prime factorization is: $\! 1100 = 2^2 \cdot 5^2 \cdot 11^1 = 4 \cdot 25 \cdot 11$

Quantity Value Explanation
Total number of groups up to isomorphism 51
Number of abelian groups up to isomorphism 4 (number of abelian groups of order $2^2$) $\times$ (number of abelian groups of order $5^2$) $\times$ (number of abelian groups of order $11^1$) = (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 1) = $2 \times 2 \times 1 = 4$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 4 (number of groups of order 4) $\times$ (number of groups of order 25) $\times$ (number of groups of order 11) = $2 \times 2 \times 1 = 4$. 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.
Note that, in this case, the nilpotent groups are the same as the abelian groups, which follows from the fact that abelian implies nilpotent and nilpotent of cube-free order implies abelian.
Number of supersolvable groups up to isomorphism 51 All groups of the order are supersolvable.
Number of solvable groups up to isomorphism 51 All groups of the order are solvable.
Number of simple non-abelian groups up to isomorphism 0 Follows from all groups of the order being solvable.

## GAP implementation

The order 1100 is part of GAP's SmallGroup library. Hence, any group of order 1100 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 1100 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(1100);

There are 51 groups of order 1100.
They are sorted by their Frattini factors.
1 has Frattini factor [ 110, 1 ].
2 has Frattini factor [ 110, 2 ].
3 has Frattini factor [ 110, 3 ].
4 has Frattini factor [ 110, 4 ].
5 has Frattini factor [ 110, 5 ].
6 has Frattini factor [ 110, 6 ].
7 has Frattini factor [ 220, 7 ].
8 has Frattini factor [ 220, 8 ].
9 has Frattini factor [ 220, 9 ].
10 has Frattini factor [ 220, 10 ].
11 has Frattini factor [ 220, 11 ].
12 has Frattini factor [ 220, 12 ].
13 has Frattini factor [ 220, 13 ].
14 has Frattini factor [ 220, 14 ].
15 has Frattini factor [ 220, 15 ].
16 has Frattini factor [ 550, 7 ].
17 has Frattini factor [ 550, 8 ].
18 has Frattini factor [ 550, 9 ].
19 has Frattini factor [ 550, 10 ].
20 has Frattini factor [ 550, 11 ].
21 has Frattini factor [ 550, 12 ].
22 has Frattini factor [ 550, 13 ].
23 has Frattini factor [ 550, 14 ].
24 has Frattini factor [ 550, 15 ].
25 has Frattini factor [ 550, 16 ].
26 - 51 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.