Finite Group DatabasebetaFinite Group Databasebeta| order | 666 |
| isoclinism | isoclinic class of C3⋊D111 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊D111 |
| unambiguous semidirect product | C37⋊(C3⋊D3) |
| Fitting, socle, derived, Hall | C32×C37 ⋊ C2 |
| Sylow | C37 ⋊ C3⋊D3 |
| Sylow | C32 ⋊ D37 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C37 | C2 |
| LCS | G ▷ C32×C37 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| transitive degree | μt(G) = 333 |
Polycyclic presentation:
2,3^2,37|b^a=b^2,c^a=c^2,d^a=d^36168 conjugacy classes — view the full character table.
| maximal subgroup of | #13050652, C3⋊D3×D37, D3×D111, … |
| maximal normal subgroup of | #13050652, C3⋊D3×D37, D3×D111, … |
| maximal quotient of | C3⋊Dic111, C2×C3⋊D111, C3⋊D333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(666, 17);