Finite Group DatabasebetaFinite Group Databasebeta| order | 35322 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊((C29⋊C7)⋊D29) |
| unambiguous semidirect product | (C29⋊C7)⋊D87 |
| Fitting, socle, derived, Hall | C3×C292 ⋊ C2×C7 |
| Hall | C3×C29×C29⋊C7 ⋊ C2 |
| Hall | C29×C29⋊C7 ⋊ D3 |
| Hall | C3⋊(C29⋊D29) ⋊ C7 |
| Sylow | C292 ⋊ D3×C7 |
| Sylow | C3 ⋊ (C29⋊C7)⋊D29 |
| 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)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61);(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)(33,34,36,38,40,42,44,46,48,50,52,54,56,58,60,61,59,57,55,53,51,49,47,45,43,41,39,37,35)Polycyclic presentation:
2,7,3,29^2|c^a=c^2,d^a=d^28,e^a=e^28,d^b=d^16489 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(35322, 38);