Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3×C5 |
| unambiguous semidirect product | C4⋊(C3⋊(C3⋊(C5⋊D5))) |
| Fitting | C32×C4×C52 ⋊ C2 |
| Hall | C32×C52 ⋊ D4 |
| Sylow | C52 ⋊ C4⋊(C3⋊D3) |
| Sylow | C32 ⋊ C4⋊(C5⋊D5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊(C5⋊D5)) |
| commutator, socle, cosocle | C2×C32×C52 | C22 |
| Fitting | C32×C4×C52 | C2 |
| hypercenter | C4 | C3⋊(C3⋊(C5⋊D5)) |
| hypocenter | C32×C52 | D4 |
| LCS | G ▷ C2×C32×C52 ▷ C32×C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
384 conjugacy classes of subgroups.
| transitive degree | μt(G) = 900 |
Polycyclic presentation:
2^3,3^2,5^2|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^4,g^a=g^4453 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 549);