# Groups of order 400

## Contents

See pages on algebraic structures of order 400| See pages on groups of a particular order

## Statistics at a glance

The number 400 has prime factors 2 and 5. The prime factorization is as follows: $\! 400 = 2^4 \cdot 5^2 = 16 \cdot 25$

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 221
Number of abelian groups up to isomorphism 10 (number of abelian groups of order $2^4$) $\times$ (number of abelian groups of order $5^2$) = (number of unordered integer partitions of 4) $\times$ (number of unordered integer partitions of 2) = $5 \times 2 = 10$.
See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 28 (number of groups of order 16) $\times$ (number of groups of order 25 -- see also groups of prime-square order) = $14 \times 2 = 28$.
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 207
Number of solvable groups up to isomorphism 221 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.

## GAP implementation

The order 400 is part of GAP's SmallGroup library. Hence, any group of order 400 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 400 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(400);

There are 221 groups of order 400.
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 - 19 have Frattini factor [ 20, 4 ].
20 - 27 have Frattini factor [ 20, 5 ].
28 - 34 have Frattini factor [ 40, 12 ].
35 - 44 have Frattini factor [ 40, 13 ].
45 - 48 have Frattini factor [ 40, 14 ].
49 has Frattini factor [ 50, 3 ].
50 has Frattini factor [ 50, 4 ].
51 has Frattini factor [ 50, 5 ].
52 has Frattini factor [ 80, 49 ].
53 has Frattini factor [ 80, 50 ].
54 has Frattini factor [ 80, 51 ].
55 has Frattini factor [ 80, 52 ].
56 has Frattini factor [ 100, 9 ].
57 has Frattini factor [ 100, 10 ].
58 has Frattini factor [ 100, 11 ].
59 has Frattini factor [ 100, 12 ].
60 - 75 have Frattini factor [ 100, 13 ].
76 - 91 have Frattini factor [ 100, 14 ].
92 - 107 have Frattini factor [ 100, 15 ].
108 - 115 have Frattini factor [ 100, 16 ].
116 has Frattini factor [ 200, 40 ].
117 - 123 have Frattini factor [ 200, 41 ].
124 - 128 have Frattini factor [ 200, 42 ].
129 - 133 have Frattini factor [ 200, 43 ].
134 has Frattini factor [ 200, 44 ].
135 - 141 have Frattini factor [ 200, 45 ].
142 - 148 have Frattini factor [ 200, 46 ].
149 - 155 have Frattini factor [ 200, 47 ].
156 - 162 have Frattini factor [ 200, 48 ].
163 - 180 have Frattini factor [ 200, 49 ].
181 - 190 have Frattini factor [ 200, 50 ].
191 - 200 have Frattini factor [ 200, 51 ].
201 - 204 have Frattini factor [ 200, 52 ].
205 - 221 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.