Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C5 |
| direct product | C7×C3⋊((C5⋊D5)⋊ΓL(1,𝔽8)) |
| unambiguous semidirect product | C3⋊(C7×(C5⋊D5)⋊ΓL(1,𝔽8)) |
| unambiguous semidirect product | (C3⋊(C5⋊D5))⋊(C7×ΓL(1,𝔽8)) |
| Fitting, socle | C3×C52×C72 ⋊ C2×C3 |
| derived | C3×C52×C7 ⋊ C2×C3×C7 |
| Hall | C3×C7×C7⋊(C52⋊C3) ⋊ C2 |
| Hall | C52×C72 ⋊ C3×D3 |
| Sylow | C72 ⋊ C3⋊((C5⋊D5)⋊C3) |
| Sylow | C52 ⋊ D3×C7×ΓL(1,𝔽8) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C3⋊((C5⋊D5)⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C3×C52×C7 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C3×C52×C7 | |
| UCS | C1 ◁ C7 | |
168 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 315 |
693 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 445);