Finite Group DatabasebetaFinite Group Databasebeta| order | 666 |
| isoclinism | isoclinic class of C37⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×C3×C37⋊C3 |
| unambiguous semidirect product | (C2×C37)⋊C32 |
| unambiguous semidirect product | (C2×C3×C37)⋊C3 |
| Fitting, socle | C2×C3×C37 ⋊ C3 |
| derived, Sylow | C37 ⋊ C2×C32 |
| Hall | C2×C37 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C37⋊C3 |
| commutator, cosocle, hypocenter | C37 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C37 | C3 |
| LCS | G ▷ C37 | |
| UCS | C1 ◁ C2×C3 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 222 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic37⋊C3, Dic3×C37⋊C3, C3⋊(Dic37⋊C3), … |
| maximal normal subgroup of | C3×Dic37⋊C3, Dic3×C37⋊C3, C3⋊(Dic37⋊C3), … |
| maximal quotient of | C3×C4×C37⋊C3, C22×C3×C37⋊C3, C2×C9×C37⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(666, 12);