# Groups of order 300

## Contents

See pages on algebraic structures of order 300| See pages on groups of a particular order

## Statistics at a glance

The number 300 has prime factors 2, 3, and 5. The prime factorization is:

$\! 300 = 2^2 \cdot 3^1 \cdot 5^2 = 4 \cdot 3 \cdot 25$

There are both solvable and non-solvable groups of this order. For the non-solvable groups, the only non-abelian composition factor is alternating group:A5 (order 60), hence the other composition factor must be cyclic group:Z5. In fact, the only non-solvable group is direct product of A5 and Z5.

Quantity Value Explanation
Number of groups up to isomorphism 49
Number of abelian groups up to isomorphism 4 (number of abelian groups of order $2^2$) $\times$ (number of abelian groups of order $3^1$) $\times$ (number of abelian groups of order $5^2$) = $2 \times 1 \times 2 = 4$.
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 3) $\times$ (number of groups of order 25) = $2 \times 1 \times 2 = 4$.
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.
Number of supersolvable groups up to isomorphism 36
Number of solvable groups up to isomorphism 48 The only non-solvable group of this order is direct product of A5 and Z5.

## GAP implementation

The order 300 is part of GAP's SmallGroup library. Hence, any group of order 300 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 300 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(300);

There are 49 groups of order 300.
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 has Frattini factor [ 60, 8 ].
8 has Frattini factor [ 60, 9 ].
9 has Frattini factor [ 60, 10 ].
10 has Frattini factor [ 60, 11 ].
11 has Frattini factor [ 60, 12 ].
12 has Frattini factor [ 60, 13 ].
13 has Frattini factor [ 150, 5 ].
14 has Frattini factor [ 150, 6 ].
15 has Frattini factor [ 150, 7 ].
16 has Frattini factor [ 150, 8 ].
17 has Frattini factor [ 150, 9 ].
18 has Frattini factor [ 150, 10 ].
19 has Frattini factor [ 150, 11 ].
20 has Frattini factor [ 150, 12 ].
21 has Frattini factor [ 150, 13 ].
22 - 49 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.