Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of C3⋊(C5⋊S4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×C3⋊(C5⋊S4) |
| unambiguous semidirect product | C3⋊(C5×C5⋊S4) |
| unambiguous semidirect product | C5⋊(C5×C3⋊S4) |
| unambiguous semidirect product | A4⋊(C5×D15) |
| Fitting, socle | C22×C3×C52 ⋊ D3 |
| derived | C3×C5×A4 ⋊ C2×C5 |
| Sylow | C52 ⋊ C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C3⋊(C5⋊S4) |
| commutator, cosocle, hypocenter | C3×C5×A4 | C2×C5 |
| Frattini | C1 | G |
| Fitting, socle | C22×C3×C52 | D3 |
| LCS | G ▷ C3×C5×A4 | |
| UCS | C1 ◁ C5 | |
148 conjugacy classes of subgroups.
| transitive degree | μt(G) = 120 |
165 conjugacy classes — view the full character table.
| maximal subgroup of | C5×C5⋊(A4⋊S4), #72260302, C5×D5×C3⋊S4 |
| maximal normal subgroup of | #72260302, C5×D5×C3⋊S4 |
| maximal quotient of | C5×C5⋊(A4⋊S4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 689);