Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C9 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×C27⋊C3 |
| unambiguous semidirect product | (C2×C4×C27)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C32 |
| commutator | C3 | C2×C3×C4×C9 |
| Frattini, cosocle | C2×C9 | C22×C32 |
| socle | C22×C3 | C2×C3×C9 |
| prime radical | C22×C32 | C2×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2)(7,8,11,9,13,19,15,22,27,10,14,18,16,23,26,24,30,32,17,12,20,25,21,28,31,29,33);(3,4,5,6)(8,12,14)(11,18,20)(13,21,23)(19,26,28)(22,29,30)(27,32,33)264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 209);