Finite Group DatabasebetaFinite Group Databasebeta| order | 32490 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C3×C19 |
| unambiguous semidirect product | C3⋊(C3⋊((C19⋊D19)⋊C5)) |
| unambiguous semidirect product | (C3⋊(C3⋊(C19⋊D19)))⋊C5 |
| Fitting, socle, derived, Hall | C32×C192 ⋊ C2×C5 |
| Hall | C32×C192⋊C5 ⋊ C2 |
| Hall | C3⋊(C3⋊(C19⋊D19)) ⋊ C5 |
| Hall | C192⋊C5 ⋊ C3⋊D3 |
| Sylow | C192 ⋊ C5×C3⋊D3 |
| Sylow | C32 ⋊ (C19⋊D19)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C32×C192 | C2×C5 |
| LCS | G ▷ C32×C192 | |
| UCS | C1 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 855 |
A minimal faithful permutation representation — 3 generators on 101 points:
(2,3)(5,6)(7,8,10,15,27)(9,22,23,59,77,13,12,47,49,56)(11,36,38,88,94,18,19,66,67,101)(14,51,52,93,87,25,24,76,79,84)(16,61,62,96,72,30,31,89,81,42)(17,29,64,80,43,34,35,60,54,74)(20,39,70,90,26,40,41,69,91,53)(21,71,75,82,83,44,45,68,55,98)(28,85,86,48,33,57,58,100,37,32)(46,50,65,95,99,73,63,78,92,97);(4,5,6)(7,9,14,26,54,81,97,98,101,86,100,94,83,99,96,80,53,25,13)(8,11,20,42,73,51,37,12,21,43,74,44,22,48,24,46,72,40,18)(10,16,32,34,23,50,19,39,52,71,45,76,41,36,63,47,17,33,30)(15,28,59,66,93,68,69,78,29,31,61,35,65,70,75,79,38,49,57)(27,55,62,84,85,91,60,88,77,92,95,56,67,64,90,58,87,89,82);(1,2,3)Polycyclic presentation:
2,5,3^2,19^2|c^a=c^2,d^a=d^2,e^a=e^18,f^a=f^18,e^b=e^12f^2,f^b=e^18f^11354 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32490, 151);