# Groups of order 240

See pages on algebraic structures of order 240| See pages on groups of a particular order

## Statistics at a glance

### Factorization and useful forms

The number 240 has prime factors 2, 3, and 5, and prime factorization $\! 240 = 2^4 \cdot 3^1 \cdot 5^1 = 16 \cdot 3 \cdot 5$

Other useful expressions for this number are: $\! 240 = 2(5!) = (5^2 - 1)(5^2 -5)/2$

### Group counts

Quantity Value Explanation GAP verification
Total number of abelian groups (i.e., finite abelian groups) up to isomorphism 5 (number of abelian groups of order $2^4$) times (number of abelian groups of order $3^1$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of $4$) times (number of unordered integer partitions of $1$) times (number of unordered integer partitions of $1$) = $5 \times 1 \times 1 = 5$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups. [SHOW MORE]
Total number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 14 (number of groups of order 16) times (number of groups of order 3) times (number of groups of order 5) = $14 \times 1 \times 1 = 14$. 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. [SHOW MORE]
Total number of solvable groups (i.e., finite solvable groups) up to isomorphism 200 See note on non-solvable groups [SHOW MORE]
Number of non-solvable groups up to isomorphism 8 See #Classification of non-solvable groups below [SHOW MORE]
Number of almost quasisimple groups up to isomorphism 2 double cover of symmetric group:S5 of minus type (ID: (240,89)) and double cover of symmetric group:S5 of plus type (ID: (240,90)) [SHOW MORE]

## Classification of non-solvable groups

The classification proceeds in steps, which are presented in sequence for clarity. The description is not complete, and Steps (4) and (5) need to be filled in:

Step no. What we are trying to find What we conclude Explanation
1 All the possibilities for simple non-abelian group of order dividing 240 the only simple non-abelian group is alternating group:A5, and it has order 60 This follows from A5 is the simple non-abelian group of smallest order and the fact that there is no simple non-abelian group of order 80, 120, or 240.
2 All the possibilities for the (unordered) collection of composition factors of a non-solvable group of order 240 one occurrence of alternating group:A5 and two occurrences of cyclic group:Z2 At least one of the composition factors must be simple non-abelian for the group to be non-solvable. So one slot goes to alternating group:A5. This takes up 60 of the 240, leaving $240/60 = 4$, which must be taken up by two copies of cyclic group:Z2.
3 All the possibilities for the composition series of a group of order 240 as a descending series, the composition series could have composition factors (in order): $\mathbb{Z}_2, \mathbb{Z}_2,A_5$ $\mathbb{Z}_2,A_5,\mathbb{Z}_2$ $A_5,\mathbb{Z}_2,\mathbb{Z}_2$

## GAP implementation

The order 240 is part of GAP's SmallGroup library. Hence, all groups of order 240 can be constructed using the SmallGroup function and have group IDs. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

gap> SmallGroupsInformation(240);

There are 208 groups of order 240.
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 - 31 have Frattini factor [ 60, 8 ].
32 has Frattini factor [ 60, 9 ].
33 - 48 have Frattini factor [ 60, 10 ].
49 - 64 have Frattini factor [ 60, 11 ].
65 - 80 have Frattini factor [ 60, 12 ].
81 - 88 have Frattini factor [ 60, 13 ].
89 - 91 have Frattini factor [ 120, 34 ].
92 - 94 have Frattini factor [ 120, 35 ].
95 - 101 have Frattini factor [ 120, 36 ].
102 - 104 have Frattini factor [ 120, 37 ].
105 - 107 have Frattini factor [ 120, 38 ].
108 - 110 have Frattini factor [ 120, 39 ].
111 - 117 have Frattini factor [ 120, 40 ].
118 - 124 have Frattini factor [ 120, 41 ].
125 - 151 have Frattini factor [ 120, 42 ].
152 - 154 have Frattini factor [ 120, 43 ].
155 - 164 have Frattini factor [ 120, 44 ].
165 - 174 have Frattini factor [ 120, 45 ].
175 - 184 have Frattini factor [ 120, 46 ].
185 - 188 have Frattini factor [ 120, 47 ].
189 - 208 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.