Groups of order 144

Contents

This article gives information about, and links to more details on, groups of order 144
See pages on algebraic structures of order 144| See pages on groups of a particular order

Statistics at a glance

The number 144 has prime factorization $144 = 2^4 \cdot 3^2$. 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 List/comment
Total number of groups 197
Number of abelian groups 10 ((number of abelian groups of order $2^4$) times ((number of abelian groups of order $3^2$) = (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 28 ((number of groups of order 16) times (number of groups of order 9 = $14 \times 2 = 28$. 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 solvable groups 197 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 groups 0 Follows from all groups of this order being solvable

GAP implementation

The order 144 is part of GAP's SmallGroup library. Hence, any group of order 144 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 144 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(144);

There are 197 groups of order 144.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 - 19 have Frattini factor [ 12, 4 ].
20 - 27 have Frattini factor [ 12, 5 ].
28 has Frattini factor [ 18, 3 ].
29 has Frattini factor [ 18, 4 ].
30 has Frattini factor [ 18, 5 ].
31 - 33 have Frattini factor [ 24, 12 ].
34 - 36 have Frattini factor [ 24, 13 ].
37 - 46 have Frattini factor [ 24, 14 ].
47 - 50 have Frattini factor [ 24, 15 ].
51 has Frattini factor [ 36, 9 ].
52 - 67 have Frattini factor [ 36, 10 ].
68 has Frattini factor [ 36, 11 ].
69 - 84 have Frattini factor [ 36, 12 ].
85 - 100 have Frattini factor [ 36, 13 ].
101 - 108 have Frattini factor [ 36, 14 ].
109 has Frattini factor [ 48, 48 ].
110 has Frattini factor [ 48, 49 ].
111 has Frattini factor [ 48, 50 ].
112 has Frattini factor [ 48, 51 ].
113 has Frattini factor [ 48, 52 ].
114 has Frattini factor [ 72, 39 ].
115 - 119 have Frattini factor [ 72, 40 ].
120 has Frattini factor [ 72, 41 ].
121 - 123 have Frattini factor [ 72, 42 ].
124 - 126 have Frattini factor [ 72, 43 ].
127 - 129 have Frattini factor [ 72, 44 ].
130 - 136 have Frattini factor [ 72, 45 ].
137 - 154 have Frattini factor [ 72, 46 ].
155 - 157 have Frattini factor [ 72, 47 ].
158 - 167 have Frattini factor [ 72, 48 ].
168 - 177 have Frattini factor [ 72, 49 ].
178 - 181 have Frattini factor [ 72, 50 ].
182 - 197 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.