# Groups of order 1500

## Contents

See pages on algebraic structures of order 1500| See pages on groups of a particular order

## Statistics at a glance

The order 1500 has the following prime factorization:

$\! 1500 = 2^2 \cdot 3^1 \cdot 5^3 = 4 \cdot 3 \cdot 125$

Quantity Value Explanation
Total number of groups up to isomorphism 174
Number of abelian groups up to isomorphism 6 (number of abelian groups of order $2^2$) $\times$ (number of abelian groups of order $3^1$) $\times$ (number of abelian groups of order $5^3$) = (number of unordered integer partitions of 2) $\times$ (number of unordered integer partitions of 1) $\times$ (number of unordered integer partitions of 3) = $2 \times 1 \times 3 = 6$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups up to isomorphism 10 (number of groups of order 4) $\times$ (number of groups of order 3) $\times$ (number of groups of order 125) = $2 \times 1 \times 5 = 10$. 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 128
Number of solvable groups up to isomorphism 172 the only non-solvable groups are direct product of A5 and Z25 and direct product of A5 and E25. Both of these contain alternating group:A5 as a direct factor.
Number of simple non-abelian groups up to isomorphism 0

## GAP implementation

The order 1500 is part of GAP's SmallGroup library. Hence, any group of order 1500 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 1500 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(1500);

There are 174 groups of order 1500.
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 - 19 have Frattini factor [ 150, 8 ].
20 - 21 have Frattini factor [ 150, 9 ].
22 - 24 have Frattini factor [ 150, 10 ].
25 - 28 have Frattini factor [ 150, 11 ].
29 - 30 have Frattini factor [ 150, 12 ].
31 - 33 have Frattini factor [ 150, 13 ].
34 has Frattini factor [ 300, 22 ].
35 has Frattini factor [ 300, 23 ].
36 has Frattini factor [ 300, 24 ].
37 has Frattini factor [ 300, 25 ].
38 has Frattini factor [ 300, 26 ].
39 has Frattini factor [ 300, 27 ].
40 - 43 have Frattini factor [ 300, 28 ].
44 - 46 have Frattini factor [ 300, 29 ].
47 - 48 have Frattini factor [ 300, 30 ].
49 - 50 have Frattini factor [ 300, 31 ].
51 - 54 have Frattini factor [ 300, 32 ].
55 - 57 have Frattini factor [ 300, 33 ].
58 - 59 have Frattini factor [ 300, 34 ].
60 - 61 have Frattini factor [ 300, 35 ].
62 - 63 have Frattini factor [ 300, 36 ].
64 - 67 have Frattini factor [ 300, 37 ].
68 - 69 have Frattini factor [ 300, 38 ].
70 - 72 have Frattini factor [ 300, 39 ].
73 - 74 have Frattini factor [ 300, 40 ].
75 has Frattini factor [ 300, 41 ].
76 - 78 have Frattini factor [ 300, 42 ].
79 has Frattini factor [ 300, 43 ].
80 - 83 have Frattini factor [ 300, 44 ].
84 - 85 have Frattini factor [ 300, 45 ].
86 - 88 have Frattini factor [ 300, 46 ].
89 - 92 have Frattini factor [ 300, 47 ].
93 - 94 have Frattini factor [ 300, 48 ].
95 - 97 have Frattini factor [ 300, 49 ].
98 has Frattini factor [ 750, 26 ].
99 has Frattini factor [ 750, 27 ].
100 has Frattini factor [ 750, 28 ].
101 has Frattini factor [ 750, 29 ].
102 has Frattini factor [ 750, 30 ].
103 has Frattini factor [ 750, 31 ].
104 has Frattini factor [ 750, 32 ].
105 has Frattini factor [ 750, 33 ].
106 has Frattini factor [ 750, 34 ].
107 has Frattini factor [ 750, 35 ].
108 has Frattini factor [ 750, 36 ].
109 has Frattini factor [ 750, 37 ].
110 has Frattini factor [ 750, 38 ].
111 has Frattini factor [ 750, 39 ].
112 - 174 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.