Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C5⋊D5)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3×C5 |
| direct product | C2×C3⋊(C3⋊#1039) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊#1039) |
| derived, Hall | C32×C52 ⋊ C2×C4 |
| Sylow | C52 ⋊ C2×C3⋊Dic3 |
| Sylow | C32 ⋊ C2×#1039 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊(C3⋊(C5⋊D5)) |
| commutator, hypocenter | C32×C52 | C2×C4 |
| Frattini | C2 | C2×C3⋊(C3⋊(C5⋊D5)) |
| Fitting, socle, prime radical | C22×C32×C52 | C2 |
| cosocle | C2×C32×C52 | C22 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 ◁ C22 | |
384 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1800 |
456 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3⋊D3×#1039, C2×C3⋊Dic3×C5⋊D5, C2×C4×C3⋊(C3⋊(C5⋊D5)), … |
| maximal normal subgroup of | C2×C3⋊D3×#1039, C2×C3⋊Dic3×C5⋊D5, C2×C4×C3⋊(C3⋊(C5⋊D5)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 550);