Finite Group DatabasebetaFinite Group Databasebeta| order | 32850 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C73⋊C3)⋊#3874 |
| unambiguous semidirect product | (C73⋊#3874)⋊C3 |
| Fitting, socle, Hall | C52×C73 ⋊ C3×D3 |
| derived | C73×C52⋊C3 ⋊ C2×C3 |
| Hall | C52⋊C3×C73⋊C3 ⋊ C2 |
| Sylow | C73 ⋊ C3×#3874 |
| Sylow | C52 ⋊ C3⋊(D73⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C73×C52⋊C3 | C2×C3 |
| Fitting, socle | C52×C73 | C3×D3 |
| LCS | G ▷ C73×C52⋊C3 | |
| UCS | C1 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 88 |
| transitive degree | μt(G) = 1095 |
A minimal faithful permutation representation — 2 generators on 88 points:
(1,3,6,11,15)(2,8,14,7,12,13,10,9,5,4)(17,21,22,18,19,23)(20,29,30,24,25,39)(26,42,43,32,33,56)(27,41,37,35,31,28)(34,61,62,45,46,75)(36,59,60,48,44,53)(38,40,58,50,51,54)(47,63,72,49,64,78)(52,57,77,65,66,73)(55,82,80,69,70,68)(67,76,83,71,79,87)(74,88,86,84,85,81);(1,2,4)(3,5,9)(6,10,13)(7,11,12)(8,15,14)(16,17,20,26,34,47,48,35,19,23,28,38,52,67,69,51,25,31,39,53,68,81,84,66,33,44,54,56,72,83,88,79,46,63,70,73,75,62,77,82,64,61,76,85,87,78,43,58,59,42,57,74,86,80,60,30,41,29,40,55,71,65,50,37,22,21,27,36,49,45,32,24,18)Polycyclic presentation:
2,3^2,73,5^2|c^a=c^2,d^a=d^72,e^a=ef^2,f^a=f^4,d^b=d^8,e^c=e^3f^3,f^c=e^4f171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32850, 68);