Finite Group DatabasebetaFinite Group Databasebeta| order | 22050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C5×C7 |
| unambiguous semidirect product | C5⋊(C5⋊#11725078) |
| derived, Hall | C52×C72 ⋊ C2×C9 |
| Hall | C52×#89205 ⋊ C2 |
| Hall | C5⋊(C5⋊(C7⋊D7)) ⋊ C9 |
| Hall | #89205 ⋊ C5⋊D5 |
| Sylow | C72 ⋊ C9×C5⋊D5 |
| Sylow | C52 ⋊ #11725078 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C5⋊(C5⋊#64425) |
| commutator, hypocenter | C52×C72 | C2×C9 |
| Fitting, socle, cosocle | C3×C52×C72 | C2×C3 |
| prime radical | C3×C5⋊(C5⋊(C7⋊D7)) | C3 |
| LCS | G ▷ C52×C72 | |
| UCS | C1 ◁ C3 | |
256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1575 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,5,9,3,8,4,6,2,7)(11,14,15,12,13,16)(18,22,21,19,20,23)(25,26)(27,28)(30,31)(32,33);(10,11,13,16,15,14,12)(17,18,21,22,20,23,19)(29,30,32,33,31);(24,25,27,28,26)Polycyclic presentation:
2,3,5^2,7^2,3|b^3=g,c^a=c^4,d^a=d^4,e^a=e^6,f^a=f^6,e^b=e^2,f^b=f^4726 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22050, 244);