Finite Group DatabasebetaFinite Group Databasebeta| order | 3690 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D615 |
| unambiguous semidirect product | C3⋊(C3×D205) |
| unambiguous semidirect product | C5⋊(C3×D123) |
| unambiguous semidirect product | C41⋊(C3×D15) |
| Fitting, socle, Hall | C32×C5×C41 ⋊ C2 |
| derived | C3×C5×C41 ⋊ C2×C3 |
| Hall | C32×C41 ⋊ D5 |
| Hall | C5×C41 ⋊ C3×D3 |
| Hall | C32×C5 ⋊ D41 |
| Sylow | C41 ⋊ C3×D15 |
| Sylow | C32 ⋊ D205 |
| Sylow | C5 ⋊ C3×D123 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D615 |
| commutator, cosocle, hypocenter | C3×C5×C41 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C5×C41 | C2 |
| LCS | G ▷ C3×C5×C41 | |
| UCS | C1 ◁ C3 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 1230 |
927 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×D3075, C3×C3⋊(C5⋊(D41⋊C5)), C3×C3⋊(C5⋊D205), … |
| maximal normal subgroup of | C3×C3⋊(C5⋊(D41⋊C5)), C3×C5×D615, C3×C7×D615 |
| maximal quotient of | C3×D3075, C3×C3⋊(C5⋊D205), C3×C5×D615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3690, 26);