Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C27 · Schur multiplier (H2) = C3 |
| direct product | C4×C9⋊C27 |
| unambiguous semidirect product | (C4×C9)⋊C27 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C9 | C32 |
| commutator | C3 | C3×C4×C27 |
| Frattini, cosocle | C2×C3×C9 | C2×C32 |
| socle, prime radical | C2×C32 | C2×C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C9 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 972 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,4)(6,8,11)(9,13,12)(14,15,18,16,19,24,21,26,31,17,20,25,22,27,32,29,34,37,23,28,33,30,35,38,36,39,40);(5,6,9,7,8,12,10,11,13)396 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 84);