Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(ΓL(1,𝔽8)⋊D175) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D525 |
| unambiguous semidirect product | C25⋊(C3⋊(ΓL(1,𝔽8)⋊D7)) |
| Fitting, derived | C3×C72×C25 ⋊ C2×C3 |
| Hall | C3×C7×ΓL(1,𝔽8)×C25 ⋊ C2 |
| Hall | C72×C25 ⋊ C3×D3 |
| Hall | C3×C7×ΓL(1,𝔽8) ⋊ D25 |
| Sylow | C72 ⋊ C3×D75 |
| Sylow | C25 ⋊ C3⋊(ΓL(1,𝔽8)⋊D7) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C72×C25 | C2×C3 |
| Frattini | C5 | C3⋊(C5⋊(ΓL(1,𝔽8)⋊D7)) |
| socle | C3×C5×C72 | C3×D5 |
| LCS | G ▷ C3×C72×C25 | |
| UCS | C1 | |
132 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1575 |
Polycyclic presentation:
2,3^2,5,7^2,5|d^5=g^4,c^a=c^2,d^a=d^4g,e^a=e^6,f^a=f^6,g^a=g^4,e^b=e^21317 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 196);