Finite Group DatabasebetaFinite Group Databasebeta| order | 450 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C5⋊D5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3×C5 |
| unambiguous semidirect product | C3⋊(C3⋊(C5⋊D5)) |
| unambiguous semidirect product | C5⋊(C3⋊D15) |
| Fitting, socle, derived, Hall | C32×C52 ⋊ C2 |
| Sylow | C52 ⋊ C3⋊D3 |
| Sylow | C32 ⋊ C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C52 | C2 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 225 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,3)(4,5)(7,8)(9,10)(12,13)(15,16);(6,7,9,10,8)(14,15,16);(1,2,4,5,3)(11,12,13)Polycyclic presentation:
2,3^2,5^2|b^a=b^2,c^a=c^2,d^a=d^4,e^a=e^4114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(450, 33);