Finite Group DatabasebetaFinite Group Databasebeta| order | 32634 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#24925914 |
| Fitting, derived | C3×C37×C49 ⋊ C2×C3 |
| Hall | C3×#19129581 ⋊ C2 |
| Hall | C37×C49 ⋊ C3×D3 |
| Sylow | C49 ⋊ C3⋊(D37⋊C3) |
| Sylow | C37 ⋊ D3×C49⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C3×C37×C49 | C2×C3 |
| Frattini | C7 | C3⋊#13095531 |
| socle | C3×C7×C37 | C2×ΓL(1,𝔽8) |
| 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)(42,44,48)(43,47,55)(45,51,63)(46,53,60)(49,59,57)(50,62,71)(52,66,84)(54,70,58)(56,73,72)(61,78,77)(64,80,74)(65,83,69)(67,85,87)(68,81,75)(76,79,86)(82,89,88);(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)(41,63,87,61,58,77,74,47,83,44,80,67,82,76,55,42,53,89,51,75,88,73,54,70,49,48,84,66,43,71,45,68,62,64,59,56,85,50,86,60,57,78,72,46,69,52,65,79,81)Polycyclic presentation:
2,3^2,7,37,7|d^7=f^2,c^a=c^2,e^a=e^36,d^b=d^2f,e^b=e^26,f^b=f^2939 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32634, 63);