Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of D225 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C9⋊#6014 |
| derived, Hall | C9×C25 ⋊ C8 |
| Sylow | C25 ⋊ C9⋊C8 |
| Sylow | C9 ⋊ #6014 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D225 |
| commutator, hypocenter | C9×C25 | C8 |
| Frattini | C3×C4×C5 | D15 |
| Fitting, cosocle | C4×C9×C25 | C2 |
| socle, prime radical | C2×C3×C5 | Dic15 |
| LCS | G ▷ C9×C25 | |
| UCS | C1 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1800 |
Polycyclic presentation:
2^3,3,5,3,5|a^2=b,b^2=c,d^3=f,e^5=g,d^a=d^2f^2,e^a=e^4g^4,f^a=f^2,g^a=g^4456 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 3);