Finite Group DatabasebetaFinite Group Databasebeta| order | 660 |
| isoclinism | isoclinic class of D165 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D330 |
| direct product | C2×D165 |
| unambiguous semidirect product | (C2×C3)⋊D55 |
| unambiguous semidirect product | (C2×C5)⋊D33 |
| unambiguous semidirect product | (C2×C11)⋊D15 |
| Fitting, socle | C2×C3×C5×C11 ⋊ C2 |
| derived, Hall | C3×C5×C11 ⋊ C22 |
| Hall | C5×C11 ⋊ C2×D3 |
| Hall | C3×C11 ⋊ C2×D5 |
| Hall | C3×C5 ⋊ C2×D11 |
| Sylow | C11 ⋊ C2×D15 |
| Sylow | C5 ⋊ C2×D33 |
| Sylow | C3 ⋊ C2×D55 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D165 |
| commutator, cosocle, hypocenter | C3×C5×C11 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C5×C11 | C2 |
| LCS | G ▷ C3×C5×C11 | |
| UCS | C1 ◁ C2 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 330 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21);(4,5)(7,10)(8,9)(12,21)(13,20)(14,19)(15,18)(16,17)168 conjugacy classes — view the full character table.
| maximal subgroup of | #13050357, #13050359, #13050360, … |
| maximal normal subgroup of | #13050357, #13050359, #13050360, … |
| maximal quotient of | Dic330, C4×D165, D660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(660, 39);