Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #6881 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| direct product | C4×#6882 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #732 |
| commutator | C3×C9 | C32×C4 |
| Frattini, cosocle | C2×C3×C9 | C2×C32 |
| socle | C2×C3 | C2×#732 |
| prime radical | C2×C32 | C2×H(3,𝔽3) |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C32×C4 ◁ C3×C4×C9 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,3,4)(5,6,9,8,13,19,17,12,25)(7,14,18,15,11,23,26,24,31)(10,20,27,22,29,28,21,30,16);(6,10,11,13,22,24,12,21,14)(7,15,26)(9,18,29,25,31,20,19,23,30)(16,28,27)76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 92);