Groups of order 240

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:

$$\! 240 = 2(5!) = (5^2 - 1)(5^2 -5)/2$$

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:

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.