Finite Group DatabasebetaFinite Group Databasebeta| order | 1125 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | H(3,𝔽5)⋊C9 |
| derived, Sylow | H(3,𝔽5) ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3×C5 | C52⋊C3 |
| commutator, hypocenter | H(3,𝔽5) | C9 |
| Fitting, prime radical, cosocle | C3×H(3,𝔽5) | C3 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 ◁ C3×C5 | |
17 conjugacy classes of subgroups.
| transitive degree | μt(G) = 225 |
Polycyclic presentation:
3^2,5^3|a^3=b,c^a=c^3d^2e,d^a=cde,d^c=de69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1125, 3);