Groups of order 756

This article gives information about, and links to more details on, groups of order 756
See pages on algebraic structures of order 756 | See pages on groups of a particular order

Statistics at a glance

The number 756 has prime factors 2, 3, and 7. It has the prime factorization:

$\!756=2^{2}\cdot 3^{3}\cdot 7$

All groups of this order are solvable groups, and hence finite solvable groups.

GAP implementation

The order 756 is part of GAP's SmallGroup library. Hence, any group of order 756 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 756 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(756);

  There are 189 groups of order 756.
  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 has Frattini factor [ 84, 7 ].
     8 has Frattini factor [ 84, 8 ].
     9 has Frattini factor [ 84, 9 ].
     10 has Frattini factor [ 84, 10 ].
     11 has Frattini factor [ 84, 11 ].
     12 has Frattini factor [ 84, 12 ].
     13 has Frattini factor [ 84, 13 ].
     14 has Frattini factor [ 84, 14 ].
     15 has Frattini factor [ 84, 15 ].
     16 - 21 have Frattini factor [ 126, 7 ].
     22 - 26 have Frattini factor [ 126, 8 ].
     27 - 31 have Frattini factor [ 126, 9 ].
     32 - 37 have Frattini factor [ 126, 10 ].
     38 - 40 have Frattini factor [ 126, 11 ].
     41 - 44 have Frattini factor [ 126, 12 ].
     45 - 48 have Frattini factor [ 126, 13 ].
     49 - 50 have Frattini factor [ 126, 14 ].
     51 - 52 have Frattini factor [ 126, 15 ].
     53 - 55 have Frattini factor [ 126, 16 ].
     56 - 60 have Frattini factor [ 252, 26 ].
     61 - 69 have Frattini factor [ 252, 27 ].
     70 - 75 have Frattini factor [ 252, 28 ].
     76 - 80 have Frattini factor [ 252, 29 ].
     81 - 85 have Frattini factor [ 252, 30 ].
     86 has Frattini factor [ 252, 31 ].
     87 has Frattini factor [ 252, 32 ].
     88 - 91 have Frattini factor [ 252, 33 ].
     92 - 93 have Frattini factor [ 252, 34 ].
     94 - 95 have Frattini factor [ 252, 35 ].
     96 - 98 have Frattini factor [ 252, 36 ].
     99 - 100 have Frattini factor [ 252, 37 ].
     101 - 106 have Frattini factor [ 252, 38 ].
     107 - 111 have Frattini factor [ 252, 39 ].
     112 - 117 have Frattini factor [ 252, 40 ].
     118 - 120 have Frattini factor [ 252, 41 ].
     121 - 124 have Frattini factor [ 252, 42 ].
     125 - 128 have Frattini factor [ 252, 43 ].
     129 - 130 have Frattini factor [ 252, 44 ].
     131 - 132 have Frattini factor [ 252, 45 ].
     133 - 135 have Frattini factor [ 252, 46 ].
     136 has Frattini factor [ 378, 47 ].
     137 has Frattini factor [ 378, 48 ].
     138 has Frattini factor [ 378, 49 ].
     139 has Frattini factor [ 378, 50 ].
     140 has Frattini factor [ 378, 51 ].
     141 has Frattini factor [ 378, 52 ].
     142 has Frattini factor [ 378, 53 ].
     143 has Frattini factor [ 378, 54 ].
     144 has Frattini factor [ 378, 55 ].
     145 has Frattini factor [ 378, 56 ].
     146 has Frattini factor [ 378, 57 ].
     147 has Frattini factor [ 378, 58 ].
     148 has Frattini factor [ 378, 59 ].
     149 has Frattini factor [ 378, 60 ].
     150 - 189 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.