Groups of order 168

Factorization and useful forms
The number 168 has prime factors 2, 3, and 7, with prime factorization:

$$\! 168 = 2^3 \cdot 3 \cdot 7 = 8 \cdot 3 \cdot 7$$

Other expressions for this number are:

$$\! 168 = (7^3 - 7)/2 = 2^3(2^3 - 1)(2^3 - 2) = 84(3 - 1)$$

GAP implementation
gap> SmallGroupsInformation(168);

There are 57 groups of order 168. They are sorted by their Frattini factors. 1 has Frattini factor [ 42, 1 ]. 2 has Frattini factor [ 42, 2 ]. 3 has Frattini factor [ 42, 3 ]. 4 has Frattini factor [ 42, 4 ]. 5 has Frattini factor [ 42, 5 ]. 6 has Frattini factor [ 42, 6 ]. 7 - 11 have Frattini factor [ 84, 7 ]. 12 - 18 have Frattini factor [ 84, 8 ]. 19 - 21 have Frattini factor [ 84, 9 ]. 22 has Frattini factor [ 84, 10 ]. 23 has Frattini factor [ 84, 11 ]. 24 - 28 have Frattini factor [ 84, 12 ]. 29 - 33 have Frattini factor [ 84, 13 ]. 34 - 38 have Frattini factor [ 84, 14 ]. 39 - 41 have Frattini factor [ 84, 15 ]. 42 - 57 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.