Finite Group DatabasebetaFinite Group Databasebeta| order | 32634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24925918 |
| Fitting, derived | C3×C37×C49 ⋊ C2×C3 |
| Hall | C3×#19129581 ⋊ C2 |
| Hall | C37×C49 ⋊ C3×D3 |
| Sylow | C49 ⋊ D3×C37⋊C3 |
| Sylow | C37 ⋊ C3⋊(D49⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C37×C49 | C2×C3 |
| Frattini | C7 | C3⋊#13095535 |
| socle | C3×C7×C37 | AGL(1,𝔽7) |
| LCS | G ▷ C3×C37×C49 | |
| UCS | C1 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 5439 |
A minimal faithful permutation representation — 2 generators on 89 points:
(2,3)(5,6,8)(7,10,15)(9,13,12)(11,17,24)(14,19,21)(16,22,18)(20,27,31)(23,29,28)(25,32,26)(30,35,36)(33,37,34)(38,40,39)(42,50,52,44,45,53)(43,62,64,48,49,65)(46,74,61,55,56,83)(47,79,80,59,60,89)(51,58,57,63,68,76)(54,77,86,72,73,87)(66,78,81,84,67,70)(69,82,71,85,75,88);(1,2,3)(4,5,7,11,17,25,33,38,37,22,8,12,18,26,27,10,16,23,30,32,15,21,28,34,35,19,6,9,14,20,24,31,36,39,40,29,13)(41,86,58,81,80,67,69,48,84,46,71,76,50,51,62,61,82,53,73,72,44,64,47,60,74,56,79,59,65,42,54,77,52,75,83,49,63,45,57,88,55,66,43,85,78,89,70,68,87)Polycyclic presentation:
2,3^2,7,37,7|d^7=f^6,c^a=c^2,d^a=d^6f,f^a=f^6,d^b=d^2f^3,e^b=e^26,f^b=f^2933 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 59);