Finite Group DatabasebetaFinite Group Databasebeta| order | 34164 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #19151048⋊D3 |
| Fitting, socle, derived | C3×C13×C73 ⋊ C22×C3 |
| Hall | C3×#15667865 ⋊ C22 |
| Hall | C13×C73 ⋊ C2×C3×D3 |
| Sylow | C73 ⋊ D3×D13⋊C3 |
| Sylow | C13 ⋊ D3×D73⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3×C13×C73 | C22×C3 |
| LCS | G ▷ C3×C13×C73 | |
| UCS | C1 | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 2847 |
A minimal faithful permutation representation — 2 generators on 89 points:
(2,3)(4,5,12,15,13,10)(6,11,8,16,7,14)(17,18,24)(19,28,40)(20,22,29)(21,32,23)(25,41,57)(26,42,39)(27,45,31)(30,54,36)(33,58,75)(34,59,53)(35,64,44)(37,55,38)(43,72,49)(46,47,76)(48,74,63)(50,70,61)(51,56,73)(52,60,68)(62,88,65)(66,83,78)(67,81,79)(69,77,85)(71,87,80)(82,86,89);(1,2,3)(5,6)(7,9)(8,12)(10,11)(13,15)(14,16)(18,19)(20,22)(21,25)(23,24)(26,30)(27,33)(28,36)(29,38)(31,40)(32,42)(34,43)(35,46)(37,49)(39,51)(41,52)(44,57)(45,60)(47,62)(48,53)(50,65)(54,61)(55,59)(56,69)(58,67)(63,75)(64,77)(66,79)(68,80)(70,81)(71,82)(72,78)(73,76)(74,86)(83,89)(84,85)(87,88)Polycyclic presentation:
2^2,3^2,13,73|e^a=e^12,f^a=f^72,d^b=d^2,e^b=e^12,e^c=e^3,f^c=f^64276 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34164, 256);