Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C52 · Schur multiplier (H2) = C3×C5 |
| direct product | C3×C52×ΓL(1,𝔽8)⋊D7 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3×C52×D7) |
| Fitting, socle | C3×C52×C72 ⋊ C2×C3 |
| derived, Sylow | C72 ⋊ C2×C32×C52 |
| Hall | C3×C52×C7×ΓL(1,𝔽8) ⋊ C2 |
| Hall | C52×C7⋊D7 ⋊ C32 |
| Hall | C52×C72 ⋊ C2×C32 |
| Hall | C3×C7×ΓL(1,𝔽8) ⋊ C2×C52 |
| Hall | C7⋊D7 ⋊ C32×C52 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C52 | ΓL(1,𝔽8)⋊D7 |
| commutator, cosocle, hypocenter | C72 | C2×C32×C52 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C3×C52 | |
448 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 448 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1575 |
1650 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 399);