Finite Group DatabasebetaFinite Group Databasebeta| order | 32850 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C5 |
| unambiguous semidirect product | C5⋊(C5⋊(#10620323⋊C2)) |
| unambiguous semidirect product | #10620323⋊(C5⋊D5) |
| Fitting, socle, derived, Hall | C52×C73 ⋊ C2×C9 |
| Hall | C52×#10620323 ⋊ C2 |
| Hall | C5⋊D365 ⋊ C9 |
| Hall | #10620323 ⋊ C5⋊D5 |
| Sylow | C73 ⋊ C9×C5⋊D5 |
| Sylow | C52 ⋊ #10620323⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C52×C73 | C2×C9 |
| prime radical | C5⋊(C5⋊(D73⋊C3)) | C3 |
| cosocle | C52×C73⋊C3 | C2×C3 |
| LCS | G ▷ C52×C73 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1825 |
A minimal faithful permutation representation — 3 generators on 83 points:
(2,3)(4,5)(7,8)(9,10)(12,24,40,17,28,23,21,16,26,13,19,52,15,33,18,22,14,35)(20,51,64,47,53,63,50,39,71,30,49,68,32,74,48,69,31,70)(25,66,41,44,67,65,46,60,42,37,45,43,29,54,61,34,38,27)(36,75,82,73,83,62,76,80,78,57,81,79,59,77,72,55,58,56);(6,7,9,10,8)(11,12,14,20,18,27,31,37,15,22,34,55,48,72,60,57,19,29,33,53,54,75,77,71,32,51,59,69,65,82,80,78,26,41,45,68,52,40,64,66,43,35,56,58,79,61,50,73,49,47,70,83,81,67,74,28,44,24,36,38,62,63,76,46,21,17,25,39,42,23,30,16,13);(1,2,4,5,3)Polycyclic presentation:
2,3^2,5^2,73|b^3=c,d^a=d^4,e^a=e^4,f^a=f^72,f^b=f^2,f^c=f^8226 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32850, 77);