# Groups of order 600

## Contents

See pages on algebraic structures of order 600| See pages on groups of a particular order

## Statistics at a glance

The number 600 has prime factors 2, 3, and 5. The prime factorization is: $\! 600 = 2^3 \cdot 3 \cdot 5^2 = 8 \cdot 3 \cdot 25$

There are both solvable and non-solvable groups of this order. For all the non-solvable groups, the unique non-abelian composition factor is alternating group:A5 (order 60), and the abelian composition factors are thus cyclic group:Z2 and cyclic group:Z5.

## GAP implementation

The order 600 is part of GAP's SmallGroup library. Hence, any group of order 600 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 600 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(600);

There are 205 groups of order 600.
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 - 13 have Frattini factor [ 60, 8 ].
14 has Frattini factor [ 60, 9 ].
15 - 19 have Frattini factor [ 60, 10 ].
20 - 24 have Frattini factor [ 60, 11 ].
25 - 29 have Frattini factor [ 60, 12 ].
30 - 32 have Frattini factor [ 60, 13 ].
33 has Frattini factor [ 120, 36 ].
34 has Frattini factor [ 120, 37 ].
35 has Frattini factor [ 120, 38 ].
36 has Frattini factor [ 120, 39 ].
37 has Frattini factor [ 120, 40 ].
38 has Frattini factor [ 120, 41 ].
39 has Frattini factor [ 120, 42 ].
40 has Frattini factor [ 120, 43 ].
41 has Frattini factor [ 120, 44 ].
42 has Frattini factor [ 120, 45 ].
43 has Frattini factor [ 120, 46 ].
44 has Frattini factor [ 120, 47 ].
45 has Frattini factor [ 150, 5 ].
46 has Frattini factor [ 150, 6 ].
47 has Frattini factor [ 150, 7 ].
48 has Frattini factor [ 150, 8 ].
49 has Frattini factor [ 150, 9 ].
50 has Frattini factor [ 150, 10 ].
51 has Frattini factor [ 150, 11 ].
52 has Frattini factor [ 150, 12 ].
53 has Frattini factor [ 150, 13 ].
54 has Frattini factor [ 300, 22 ].
55 has Frattini factor [ 300, 23 ].
56 has Frattini factor [ 300, 24 ].
57 - 61 have Frattini factor [ 300, 25 ].
62 - 66 have Frattini factor [ 300, 26 ].
67 - 71 have Frattini factor [ 300, 27 ].
72 has Frattini factor [ 300, 28 ].
73 has Frattini factor [ 300, 29 ].
74 has Frattini factor [ 300, 30 ].
75 has Frattini factor [ 300, 31 ].
76 has Frattini factor [ 300, 32 ].
77 has Frattini factor [ 300, 33 ].
78 has Frattini factor [ 300, 34 ].
79 has Frattini factor [ 300, 35 ].
80 - 84 have Frattini factor [ 300, 36 ].
85 - 91 have Frattini factor [ 300, 37 ].
92 - 98 have Frattini factor [ 300, 38 ].
99 - 105 have Frattini factor [ 300, 39 ].
106 - 110 have Frattini factor [ 300, 40 ].
111 - 113 have Frattini factor [ 300, 41 ].
114 has Frattini factor [ 300, 42 ].
115 has Frattini factor [ 300, 43 ].
116 - 120 have Frattini factor [ 300, 44 ].
121 - 125 have Frattini factor [ 300, 45 ].
126 - 130 have Frattini factor [ 300, 46 ].
131 - 135 have Frattini factor [ 300, 47 ].
136 - 140 have Frattini factor [ 300, 48 ].
141 - 143 have Frattini factor [ 300, 49 ].
144 - 205 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.