Finite Group DatabasebetaFinite Group Databasebeta| order | 35322 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26044458 |
| Fitting, socle, derived, Hall | C3×C292 ⋊ C2×C7 |
| Hall | C3×#19154635 ⋊ C2 |
| Hall | #19154635 ⋊ D3 |
| Hall | C3⋊(C29⋊D29) ⋊ C7 |
| Sylow | C292 ⋊ D3×C7 |
| Sylow | C3 ⋊ #26044458 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C292 | C2×C7 |
| LCS | G ▷ C3×C292 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 609 |
A minimal faithful permutation representation — 2 generators on 61 points:
(2,3)(5,9,10,24,17,25,31,6,7,14,16,23,29,32)(8,21,22,26,27,18,19,12,13,30,15,28,11,20)(34,38,39,50,53,57,51,35,36,43,45,59,52,44)(37,47,48,61,42,54,58,40,41,55,56,46,49,60);(1,2,3)(4,5,8,13,23,14,25,16,28,32,30,19,18,9,15,26,7,11,20,22,31,27,24,29,10,17,21,12,6)(33,34,37,42,43,38,44,52,55,50,47,56,60,59,49,54,53,58,61,41,45,48,57,51,36,39,46,40,35)Polycyclic presentation:
2,7,3,29^2|c^a=c^2,d^a=d^28,e^a=e^28,d^b=d^16,e^b=e^24201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35322, 45);