Finite Group DatabasebetaFinite Group Databasebeta| order | 660 |
| isoclinism | isoclinic class of D55 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic55 |
| unambiguous semidirect product | C11⋊(C3×Dic5) |
| derived, Hall | C5×C11 ⋊ C3×C4 |
| Hall | C3×C5×C11 ⋊ C4 |
| Hall | C3×C11 ⋊ Dic5 |
| Hall | C3×C5 ⋊ Dic11 |
| Sylow | C11 ⋊ C3×Dic5 |
| Sylow | C5 ⋊ C3×Dic11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D55 |
| commutator, hypocenter | C5×C11 | C3×C4 |
| Frattini | C2 | C3×D55 |
| Fitting, socle, prime radical | C2×C3×C5×C11 | C2 |
| cosocle | C2×C5×C11 | C2×C3 |
| LCS | G ▷ C5×C11 | |
| UCS | C1 ◁ C2×C3 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 660 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,5)(6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23);(4,6,5,7)(9,12)(10,11)(14,23)(15,22)(16,21)(17,20)(18,19)174 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic5×D11, C3×D5×Dic11, C3×#89167, … |
| maximal normal subgroup of | C3×Dic5×D11, C3×D5×Dic11, C3×#89167, … |
| maximal quotient of | C3×C11⋊#168, C2×C3×Dic55, C9×Dic55, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(660, 7);