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×#6857 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | H(3,𝔽3) |
| commutator | C33 | C32×C4 |
| Frattini, cosocle | C2×C33 | C2×C32 |
| socle | C2×C32 | C2×H(3,𝔽3) |
| prime radical | C2×C3×H(3,𝔽3) | C2×C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C33×C4 ◁ G | |
138 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)(5,6,8,7,9,12,11,10,13)(15,17,19);(6,9,10)(14,15,18)(16,19,22)(17,21,20)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 67);