Finite Group DatabasebetaFinite Group Databasebeta| order | 34164 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#26015648 |
| derived | C3×C13×C73 ⋊ C3×C4 |
| Hall | C3×#15667865 ⋊ C4 |
| Hall | C13×C73 ⋊ C3×Dic3 |
| Sylow | C73 ⋊ Dic3×C13⋊C3 |
| Sylow | C13 ⋊ C3⋊(Dic73⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#19151040 |
| commutator, hypocenter | C3×C13×C73 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C13×C73 | C2×C3 |
| prime radical | C2×C3×#15667865 | C2 |
| LCS | G ▷ C3×C13×C73 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 11388 |
A minimal faithful permutation representation — 2 generators on 93 points:
(2,3)(4,5,6,7)(9,10,12)(11,14,17)(13,16,15)(18,20,19)(22,26,27,23,24,28)(25,34,35,29,30,44)(31,47,48,37,38,61)(32,46,42,40,36,33)(39,66,67,50,51,79)(41,64,65,53,49,58)(43,45,63,55,56,59)(52,71,77,57,62,80)(54,81,82,69,68,76)(60,85,84,73,74,72)(70,90,91,83,78,92)(75,93,88,86,87,89);(1,2,3)(8,9,11,10,13,17,14,18,20,12,15,19,16)(21,22,25,31,39,52,55,42,27,26,32,41,54,70,84,65,35,46,34,45,60,75,88,82,48,63,64,47,62,78,87,91,67,80,85,81,66,51,68,74,77,79,92,93,90,71,38,49,59,61,76,89,86,73,56,30,36,44,58,72,83,69,53,40,24,28,33,43,57,50,37,29,23)Polycyclic presentation:
2,3^2,13,73,2|a^2=f,c^a=c^2,e^a=e^72,d^b=d^3,e^b=e^64978 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(34164, 140);