Finite Group DatabasebetaFinite Group Databasebeta| order | 2220 |
| isoclinism | isoclinic class of D3×D37 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×D3×D37 |
| unambiguous semidirect product | (D3×C37)⋊(C2×C5) |
| unambiguous semidirect product | (C3×D37)⋊(C2×C5) |
| unambiguous semidirect product | D111⋊(C2×C5) |
| Fitting, socle, Hall | C3×C5×C37 ⋊ C22 |
| derived, Hall | C3×C37 ⋊ C22×C5 |
| Hall | C5×C37 ⋊ C2×D3 |
| Sylow | C37 ⋊ C2×C5×D3 |
| Hall | C3×C5 ⋊ C2×D37 |
| Sylow | C3 ⋊ C2×C5×D37 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D3×D37 |
| commutator, cosocle, hypocenter | C3×C37 | C22×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C37 | C22 |
| LCS | G ▷ C3×C37 | |
| UCS | C1 ◁ C5 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 555 |
300 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D3×C25×D37, C52×D3×D37, C5×D15×D37, … |
| maximal normal subgroup of | D3×C25×D37, C52×D3×D37, C5×D3×C7×D37, … |
| maximal quotient of | D3×C25×D37, C52×D3×D37, C5×D15×D37, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2220, 32);