Groups of order 972
This article gives information about, and links to more details on, groups of order 972
See pages on algebraic structures of order 972 | See pages on groups of a particular order
Statistics at a glance
The number 972 has prime factors 2 and 3. It has prime factorization:
There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
GAP implementation
The order 972 is part of GAP's SmallGroup library. Hence, any group of order 972 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 972 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(972);
There are 900 groups of order 972.
They are sorted by their Frattini factors.
1 has Frattini factor [ 6, 1 ].
2 has Frattini factor [ 6, 2 ].
3 has Frattini factor [ 12, 3 ].
4 has Frattini factor [ 12, 4 ].
5 has Frattini factor [ 12, 5 ].
6 - 44 have Frattini factor [ 18, 3 ].
45 - 64 have Frattini factor [ 18, 4 ].
65 - 93 have Frattini factor [ 18, 5 ].
94 - 98 have Frattini factor [ 36, 9 ].
99 - 115 have Frattini factor [ 36, 10 ].
116 - 186 have Frattini factor [ 36, 11 ].
187 - 225 have Frattini factor [ 36, 12 ].
226 - 245 have Frattini factor [ 36, 13 ].
246 - 274 have Frattini factor [ 36, 14 ].
275 - 316 have Frattini factor [ 54, 12 ].
317 - 363 have Frattini factor [ 54, 13 ].
364 - 373 have Frattini factor [ 54, 14 ].
374 - 403 have Frattini factor [ 54, 15 ].
404 - 411 have Frattini factor [ 108, 36 ].
412 - 417 have Frattini factor [ 108, 37 ].
418 - 447 have Frattini factor [ 108, 38 ].
448 - 469 have Frattini factor [ 108, 39 ].
470 - 482 have Frattini factor [ 108, 40 ].
483 - 591 have Frattini factor [ 108, 41 ].
592 - 633 have Frattini factor [ 108, 42 ].
634 - 680 have Frattini factor [ 108, 43 ].
681 - 690 have Frattini factor [ 108, 44 ].
691 - 720 have Frattini factor [ 108, 45 ].
721 - 727 have Frattini factor [ 162, 51 ].
728 - 740 have Frattini factor [ 162, 52 ].
741 - 747 have Frattini factor [ 162, 53 ].
748 - 750 have Frattini factor [ 162, 54 ].
751 - 756 have Frattini factor [ 162, 55 ].
757 has Frattini factor [ 324, 160 ].
758 - 764 have Frattini factor [ 324, 161 ].
765 - 770 have Frattini factor [ 324, 162 ].
771 - 774 have Frattini factor [ 324, 163 ].
775 - 777 have Frattini factor [ 324, 164 ].
778 - 784 have Frattini factor [ 324, 165 ].
785 - 795 have Frattini factor [ 324, 166 ].
796 - 803 have Frattini factor [ 324, 167 ].
804 - 807 have Frattini factor [ 324, 168 ].
808 - 812 have Frattini factor [ 324, 169 ].
813 - 819 have Frattini factor [ 324, 170 ].
820 - 834 have Frattini factor [ 324, 171 ].
835 - 841 have Frattini factor [ 324, 172 ].
842 - 854 have Frattini factor [ 324, 173 ].
855 - 861 have Frattini factor [ 324, 174 ].
862 - 864 have Frattini factor [ 324, 175 ].
865 - 870 have Frattini factor [ 324, 176 ].
871 has Frattini factor [ 486, 256 ].
872 has Frattini factor [ 486, 257 ].
873 has Frattini factor [ 486, 258 ].
874 has Frattini factor [ 486, 259 ].
875 has Frattini factor [ 486, 260 ].
876 has Frattini factor [ 486, 261 ].
877 - 900 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.