Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C52 · Schur multiplier (H2) = C23×C5 |
| direct product | C22×C3×C52×D3 |
| unambiguous semidirect product | C3⋊(C23×C3×C52) |
| unambiguous semidirect product | (C2×C3)⋊(C22×C3×C52) |
| Fitting, socle | C22×C32×C52 ⋊ C2 |
| derived | C3 ⋊ C23×C3×C52 |
| Hall | C32×C52 ⋊ C23 |
| Sylow | C32 ⋊ C23×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C3×C52 | D3 |
| commutator, cosocle, hypocenter | C3 | C23×C3×C52 |
| Frattini | C1 | G |
| Fitting, socle | C22×C32×C52 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C3×C52 | |
552 conjugacy classes of subgroups.
| transitive degree | μt(G) = 600 |
900 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C22×C3×C5×D3×D5, C22×C3×D3×C5⋊D5, C2×C3×C52×#67, … |
| maximal normal subgroup of | C22×C3×C5×D3×D5, C22×C3×D3×C5⋊D5, C2×C3×C52×#67, … |
| maximal quotient of | C2×C3×C52×#67, C22×C52×AGL(1,ℤ/9) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 746);