Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3×C5 |
| direct product | C3×Q8⋊(C52⋊C3) |
| unambiguous semidirect product | (C3×Q8)⋊(C52⋊C3) |
| Fitting | C3×C52×Q8 ⋊ C3 |
| derived, Hall | C52×Q8 ⋊ C32 |
| Sylow | C52 ⋊ C3×SL(2,𝔽3) |
| Sylow | Q8 ⋊ C3×C52⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C22⋊(C52⋊C3) |
| commutator, cosocle, hypocenter | C52×Q8 | C32 |
| Frattini | C2 | C3×C22⋊(C52⋊C3) |
| Fitting | C3×C52×Q8 | C3 |
| socle | C2×C3×C52 | A4 |
| LCS | G ▷ C52×Q8 | |
| UCS | C1 ◁ C2×C3 | |
64 conjugacy classes of subgroups.
| transitive degree | μt(G) = 360 |
141 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 511);