Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C25⋊#64418) |
| unambiguous semidirect product | C25⋊(C3⋊#64418) |
| Fitting | C3×C72×C25 ⋊ D3 |
| derived, Hall | C3×C25×#3862 ⋊ C2 |
| Hall | C72×C25 ⋊ C3⋊D3 |
| Hall | C3×#3862 ⋊ D25 |
| Sylow | C72 ⋊ C3⋊D75 |
| Sylow | C25 ⋊ C3⋊#64418 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C25×#3862 | C2 |
| Frattini | C5 | C3⋊(C5⋊#64418) |
| Fitting | C3×C72×C25 | D3 |
| socle | C3×C5×C72 | D15 |
| LCS | G ▷ C3×C25×#3862 | |
| UCS | C1 | |
114 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 25 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:
2,3^2,5,7^2,5|d^5=g^4,b^a=b^2,c^a=c^2,d^a=d^4g,e^a=f,f^a=e,g^a=g^4,e^b=e^4,f^b=f^2723 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 200);