Groups of order 840

Statistics at a glance
The number 840 has the prime factorization:

$$\! 840 = 2^3 \cdot 3^1 \cdot 5^1 \cdot 7^1 = 8 \cdot 3 \cdot 5 \cdot 7$$

There are both solvable and non-solvable groups of this order. There are two possibilities for the composition factors for non-solvable groups:


 * Non-abelian composition factor alternating group:A5 (order 60) and the remaining composition factors are cyclic group:Z2 and cyclic group:Z7.
 * Non-abelian composition factor projective special linear group:PSL(3,2) (order 168) and the other composition factor is cyclic group:Z5.

GAP implementation
gap> SmallGroupsInformation(840);

There are 186 groups of order 840. They are sorted by their Frattini factors. 1 has Frattini factor [ 210, 1 ]. 2 has Frattini factor [ 210, 2 ]. 3 has Frattini factor [ 210, 3 ]. 4 has Frattini factor [ 210, 4 ]. 5 has Frattini factor [ 210, 5 ]. 6 has Frattini factor [ 210, 6 ]. 7 has Frattini factor [ 210, 7 ]. 8 has Frattini factor [ 210, 8 ]. 9 has Frattini factor [ 210, 9 ]. 10 has Frattini factor [ 210, 10 ]. 11 has Frattini factor [ 210, 11 ]. 12 has Frattini factor [ 210, 12 ]. 13 has Frattini factor [ 420, 13 ]. 14 has Frattini factor [ 420, 14 ]. 15 has Frattini factor [ 420, 15 ]. 16 - 22 have Frattini factor [ 420, 16 ]. 23 - 27 have Frattini factor [ 420, 17 ]. 28 - 32 have Frattini factor [ 420, 18 ]. 33 - 37 have Frattini factor [ 420, 19 ]. 38 has Frattini factor [ 420, 20 ]. 39 has Frattini factor [ 420, 21 ]. 40 has Frattini factor [ 420, 22 ]. 41 has Frattini factor [ 420, 23 ]. 42 - 48 have Frattini factor [ 420, 24 ]. 49 - 55 have Frattini factor [ 420, 25 ]. 56 - 62 have Frattini factor [ 420, 26 ]. 63 - 69 have Frattini factor [ 420, 27 ]. 70 - 76 have Frattini factor [ 420, 28 ]. 77 - 83 have Frattini factor [ 420, 29 ]. 84 - 90 have Frattini factor [ 420, 30 ]. 91 - 93 have Frattini factor [ 420, 31 ]. 94 has Frattini factor [ 420, 32 ]. 95 has Frattini factor [ 420, 33 ]. 96 - 100 have Frattini factor [ 420, 34 ]. 101 - 105 have Frattini factor [ 420, 35 ]. 106 - 110 have Frattini factor [ 420, 36 ]. 111 - 115 have Frattini factor [ 420, 37 ]. 116 - 120 have Frattini factor [ 420, 38 ]. 121 - 125 have Frattini factor [ 420, 39 ]. 126 - 130 have Frattini factor [ 420, 40 ]. 131 - 133 have Frattini factor [ 420, 41 ]. 134 - 186 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.