Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C9 · Schur multiplier (H2) = C32 |
| direct product | C4×H(3,𝔽3)⋊C9 |
| unambiguous semidirect product | (C4×H(3,𝔽3))⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | H(3,𝔽3) |
| commutator | C32 | C3×C4×C9 |
| Frattini, cosocle | C2×C33 | C2×C32 |
| socle | C2×C32 | C2×H(3,𝔽3) |
| prime radical | C2×C3×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C33×C4 ◁ G | |
141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,3,4)(5,6,8,7,9,11,10,12,13)(14,15,18,16,20,19,17,21,22);(14,15,19)(16,20,22)(17,21,18)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 80);