Finite Group DatabasebetaFinite Group Databasebeta| order | 34164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26015682 |
| Fitting, socle, derived | C3×C13×C73 ⋊ C3×C4 |
| Hall | C3×#15667865 ⋊ C4 |
| Hall | C13×C73 ⋊ C3×Dic3 |
| Sylow | C73 ⋊ Dic3×C13⋊C3 |
| Sylow | C13 ⋊ C3⋊(#64408⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C13×C73 | C3×C4 |
| prime radical | C3×#19151040 | C2 |
| cosocle | C3×C13×D73 | C2×C3 |
| LCS | G ▷ C3×C13×C73 | |
| UCS | C1 | |
56 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)(5,6,8)(7,10,13)(9,12,11)(14,16,15)(18,23,28,24,20,33,21,34,26,19,27,32)(22,43,68,44,30,66,31,54,45,29,46,77)(25,55,79,36,38,75,39,53,56,37,57,64)(35,58,87,72,50,73,51,81,71,49,74,48)(40,80,88,67,60,78,61,85,83,59,82,89)(41,70,42,76,47,84,63,86,65,52,69,62);(1,2,3)(4,5,7,6,9,13,10,14,16,8,11,15,12)(17,18,22,32,48,56,77,65,26,27,41,64,74,83,89,53,45,69,46,44,67,86,76,51,34,43,37,19,25,40,62,60,81,57,80,71,49,72,87,85,38,58,82,84,61,39,24,36,54,23,35,52,70,59,29,30,47,68,55,78,88,50,75,63,20,28,42,66,79,73,33,31,21)Polycyclic presentation:
2,3,2,3,13,73|a^2=c,d^a=d^2,f^a=f^27,e^b=e^3,f^b=f^64,f^c=f^72276 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34164, 267);