Finite Group DatabasebetaFinite Group Databasebeta| order | 450 |
| isoclinism | isoclinic class of C3⋊((C5⋊D5)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊((C5⋊D5)⋊C3) |
| unambiguous semidirect product | (C3⋊(C5⋊D5))⋊C3 |
| Fitting, socle, derived | C3×C52 ⋊ C2×C3 |
| Hall | C3×C52⋊C3 ⋊ C2 |
| Sylow | C52 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C52 | C2×C3 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 45 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5,7)(6,9,13,10,8,15)(11,16,18,17,14,12);(1,2,3)(4,6,11,17,10)(5,8,14,16,9)(7,12,13,15,18)Polycyclic presentation:
2,3^2,5^2|c^a=c^2,d^a=d^4,e^a=e^4,d^b=d^3e^2,e^b=de21 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(450, 24);