Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3×C5 |
| unambiguous semidirect product | C3⋊(C5⋊(C5⋊#64418)) |
| unambiguous semidirect product | C5⋊(C3⋊(C5⋊#64418)) |
| Fitting, socle | C3×C52×C72 ⋊ D3 |
| derived, Hall | C3×C52×#3862 ⋊ C2 |
| Hall | C52×C72 ⋊ C3⋊D3 |
| Hall | C3×#3862 ⋊ C5⋊D5 |
| Sylow | C72 ⋊ C3⋊(C3⋊(C5⋊D5)) |
| Sylow | C52 ⋊ C3⋊#64418 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C52×#3862 | C2 |
| Fitting, socle | C3×C52×C72 | D3 |
| LCS | G ▷ C3×C52×#3862 | |
| UCS | C1 | |
304 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1050 |
Polycyclic presentation:
2,3^2,5^2,7^2|b^a=b^2,c^a=c^2,d^a=d^4,e^a=e^4,f^a=g,g^a=f,f^b=f^4,g^b=g^2723 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 479);