Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C4×#6861 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | H(3,𝔽3) |
| commutator | C33 | C32×C4 |
| Frattini, prime radical, cosocle | C2×C33 | C2×C32 |
| socle | C2×C32 | C2×H(3,𝔽3) |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C33×C4 ◁ G | |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 58 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 2 generators on 58 points:
(1,2,3,4)(5,6,9)(7,13,17)(8,12,18)(10,19,15)(11,22,16)(14,24,27)(20,30,31)(21,29,26)(23,28,25)(32,33,36,35,38,45,43,49,55)(34,39,50,42,37,46,41,48,44)(40,47,54,52,57,58,53,51,56);(5,7,14,8,15,25,16,26,31)(6,10,20,12,21,24,11,13,23)(9,17,27,18,19,28,22,29,30)(32,34,40)(33,37,47)(35,42,52)(36,44,54)(38,48,57)(39,51,49)(41,53,43)(45,50,58)(46,56,55)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 71);