Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C3×ΓL(1,𝔽8)⋊D175 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3×D175) |
| unambiguous semidirect product | C25⋊(C3×ΓL(1,𝔽8)⋊D7) |
| Fitting | C3×C72×C25 ⋊ C2×C3 |
| derived, Hall | C72×C25 ⋊ C2×C32 |
| Hall | C3×C7×ΓL(1,𝔽8)×C25 ⋊ C2 |
| Hall | C7⋊D175 ⋊ C32 |
| Hall | C3×C7×ΓL(1,𝔽8) ⋊ D25 |
| Sylow | C72 ⋊ C32×D25 |
| Sylow | C25 ⋊ C3×ΓL(1,𝔽8)⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ΓL(1,𝔽8)⋊D175 |
| commutator, cosocle, hypocenter | C72×C25 | C2×C32 |
| Frattini | C5 | C3×C5⋊(ΓL(1,𝔽8)⋊D7) |
| Fitting | C3×C72×C25 | C2×C3 |
| socle | C3×C5×C72 | C3×D5 |
| LCS | G ▷ C72×C25 | |
| UCS | C1 ◁ C3 | |
168 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1575 |
1326 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 174);