Finite Group DatabasebetaFinite Group Databasebeta| order | 1110 |
| isoclinism | isoclinic class of D111 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×D111 |
| unambiguous semidirect product | C3⋊(C5×D37) |
| unambiguous semidirect product | C37⋊(C5×D3) |
| Fitting, socle, Hall | C3×C5×C37 ⋊ C2 |
| derived, Hall | C3×C37 ⋊ C2×C5 |
| Hall | C5×C37 ⋊ D3 |
| Sylow | C37 ⋊ C5×D3 |
| Hall | C3×C5 ⋊ D37 |
| Sylow | C3 ⋊ C5×D37 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | D111 |
| commutator, cosocle, hypocenter | C3×C37 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C3×C5×C37 | C2 |
| LCS | G ▷ C3×C37 | |
| UCS | C1 ◁ C5 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 555 |
285 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C5×D3×D37, D5×D111, D15⋊D37, … |
| maximal normal subgroup of | C5×D3×D37, D5×D111, D15⋊D37, … |
| maximal quotient of | C5×Dic111, C2×C5×D111, C5×D333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1110, 9);