Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3×C5 |
| direct product | C3×(C5⋊D5)⋊(C49⋊C3) |
| unambiguous semidirect product | (C3×C5⋊D5)⋊(C49⋊C3) |
| Fitting | C3×C52×C49 ⋊ C2×C3 |
| derived, Hall | C52×C49 ⋊ C2×C32 |
| Hall | C3×C52⋊(C49⋊C3) ⋊ C2 |
| Hall | C49×C5⋊D5 ⋊ C32 |
| Hall | C5⋊D5 ⋊ C3×C49⋊C3 |
| Sylow | C49 ⋊ C3×(C5⋊D5)⋊C3 |
| Sylow | C52 ⋊ C2×C3×C49⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C5⋊D5)⋊(C49⋊C3) |
| commutator, cosocle, hypocenter | C52×C49 | C2×C32 |
| Frattini | C7 | C3×(C5⋊D5)⋊ΓL(1,𝔽8) |
| Fitting | C3×C52×C49 | C2×C3 |
| socle | C3×C52×C7 | C2×ΓL(1,𝔽8) |
| LCS | G ▷ C52×C49 | |
| UCS | C1 ◁ C3 | |
96 conjugacy classes of subgroups.
| transitive degree | μt(G) = 2205 |
702 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 111);