Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C3⋊#20301295 |
| Fitting, socle, derived | C3×C52×C72 ⋊ C2×C3 |
| Hall | C3×C7×C7⋊(C52⋊C3) ⋊ C2 |
| Hall | C52×C72 ⋊ C3×D3 |
| Sylow | C72 ⋊ C3⋊((C5⋊D5)⋊C3) |
| Sylow | C52 ⋊ C3⋊(ΓL(1,𝔽8)⋊D7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C52×C72 | C2×C3 |
| LCS | G ▷ C3×C52×C72 | |
| UCS | C1 | |
168 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 315 |
Polycyclic presentation:
2,3^2,5^2,7^2|c^a=c^2,d^a=d^4,e^a=e^4,f^a=f^6,g^a=g^6,d^b=de,e^b=d^2e^3,f^b=f^2645 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 447);