Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of C3⋊(C5⋊S5) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C3⋊(C5⋊S5) |
| unambiguous semidirect product | C5⋊ΓL(2,𝔽4) |
| unambiguous semidirect product | A5⋊D15 |
| Fitting | C3×C5 ⋊ S5 |
| socle, derived | C3×C5×A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | C3×C5×A5 | C2 |
| Fitting | C3×C5 | S5 |
| LCS | G ▷ C3×C5×A5 | |
| UCS | C1 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 75 |
Permutation generators:
(3,4)(7,8)(9,10)(11,13);(1,4,5,3,2)(6,8,10,9,7)(11,12,13)42 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 558);