Finite Group DatabasebetaFinite Group Databasebeta| order | 32634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24925920 |
| Fitting, derived | C3×C37×C49 ⋊ C2×C3 |
| Hall | C3×#19129581 ⋊ C2 |
| Hall | C37×C49 ⋊ C3×D3 |
| Sylow | C49 ⋊ C3⋊(D37⋊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⋊#13095537 |
| 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)(5,9,10,6,7,11)(8,17,18,12,13,26)(14,29,30,20,21,37)(15,28,24,22,19,16)(23,39,40,32,31,35)(25,27,38,34,33,36)(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)(4,5,8,14,21,31,36,37,34,24,10,9,15,23,33,13,19,26,35,40,18,28,17,27,32,22,7,11,16,25,30,38,39,29,20,12,6)(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,e^a=e^36,f^a=f^6,d^b=d^2f^3,e^b=e^26,f^b=f^2915 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 57);