Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C4 · Schur multiplier (H2) = C9 |
| direct product | C4×#6898 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C33 |
| commutator | C32 | C33×C4 |
| Frattini, socle, cosocle | C2×C32 | C2×C33 |
| prime radical | C2×C33 | C2×C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ G | |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 58 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 3 generators on 58 points:
(1,2,3,4)(5,6,9)(7,11,17)(8,12,16)(10,18,15)(13,20,25)(14,21,24)(19,26,23)(22,28,30)(27,31,29)(32,33,36)(34,37,43)(35,39,44)(38,48,42)(40,45,51)(41,47,52)(46,55,50)(49,54,57)(53,58,56);(6,10,12)(9,16,18)(11,19,21)(17,24,26)(20,27,28)(25,30,31)(32,34,40,35,41,49,42,50,56)(33,37,45,39,47,54,38,46,53)(36,43,51,44,52,57,48,55,58);(5,7,13,15,23,29,8,14,22)(6,11,20,10,19,27,12,21,28)(9,17,25,18,26,31,16,24,30)(33,38,39)(36,44,48)(37,46,47)(43,52,55)(45,53,54)(51,57,58)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 388);