Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of C5⋊D45 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C9⋊#6029 |
| derived, Hall | C52×C9 ⋊ C8 |
| Sylow | C52 ⋊ C9⋊C8 |
| Sylow | C9 ⋊ #6029 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C5⋊D45 |
| commutator, hypocenter | C52×C9 | C8 |
| Frattini | C3×C4 | C3⋊(C5⋊D5) |
| Fitting, cosocle | C4×C52×C9 | C2 |
| socle, prime radical | C2×C3×C52 | Dic3 |
| LCS | G ▷ C52×C9 | |
| UCS | C1 ◁ C4 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1800 |
Polycyclic presentation:
2^3,3,5^2,3|a^2=b,b^2=c,d^3=g,d^a=d^2g^2,e^a=e^4,f^a=f^4,g^a=g^2456 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 58);