Finite Group DatabasebetaFinite Group Databasebeta| order | 34164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26015681 |
| Fitting, socle, derived | C3×C13×C73 ⋊ C3×C4 |
| Hall | C3×#15667864 ⋊ 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×#19151039 | 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)(18,23,28,24,20,33,21,32,26,19,27,40)(22,42,64,43,30,60,31,54,44,29,45,50)(25,51,75,34,36,66,37,49,52,35,53,73)(38,77,88,63,56,87,57,80,78,55,79,89)(39,67,41,81,46,82,59,83,62,58,65,61)(47,84,71,85,69,48,70,74,86,68,76,72);(1,2,3)(4,5,7,6,9,13,10,14,16,8,11,15,12)(17,18,22,32,47,36,54,59,20,28,41,49,74,56,80,73,30,46,64,43,63,67,82,72,40,60,35,19,25,38,58,78,69,75,87,68,71,86,85,89,52,76,88,81,57,37,24,34,50,33,48,61,83,55,29,44,65,45,66,77,79,84,51,62,26,27,39,42,53,70,23,31,21)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^8,f^c=f^72276 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34164, 266);