# Groups of order 352

## Contents

See pages on algebraic structures of order 352| See pages on groups of a particular order

## Statistics at a glance

The number 352 has prime factors 2 and 11, with prime factorization: $\! 352 = 2^5 \cdot 11^1 = 32 \cdot 11$

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
Total number of groups up to isomorphism 195
Number of abelian groups up to isomorphism 7 (number of abelian groups of order $2^5$) $\times$ (number of abelian groups of order $11^1$) = (number of unordered integer partitions of 5) $\times$ (number of unordered integer partitions of 1) = $7 \times 1 = 7$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 51 (number of groups of order 32) $\times$ (number of groups of order 11) = $51 \times 1 = 51$.
Number of supersolvable groups up to isomorphism 195 All groups of this order are supersolvable.
Number of solvable groups up to isomorphism 195 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 non-abelian groups of this order 0 Follows from all groups of this order being solvable.

## GAP implementation

The order 352 is part of GAP's SmallGroup library. Hence, any group of order 352 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 352 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(352);

There are 195 groups of order 352.
They are sorted by their Frattini factors.
1 has Frattini factor [ 22, 1 ].
2 has Frattini factor [ 22, 2 ].
3 - 43 have Frattini factor [ 44, 3 ].
44 - 62 have Frattini factor [ 44, 4 ].
63 - 148 have Frattini factor [ 88, 11 ].
149 - 172 have Frattini factor [ 88, 12 ].
173 - 187 have Frattini factor [ 176, 41 ].
188 - 193 have Frattini factor [ 176, 42 ].
194 - 195 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.