Finite Group DatabasebetaFinite Group Databasebeta| order | 33550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20269527 |
| Fitting, socle, derived | C5×C11×C61 ⋊ C2×C5 |
| Hall | C5×#16809822 ⋊ C2 |
| Hall | C11×C61 ⋊ C5×D5 |
| Sylow | C61 ⋊ C5⋊AGL(1,𝔽11) |
| Sylow | C11 ⋊ C5⋊(D61⋊C5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5×C11×C61 | C2×C5 |
| LCS | G ▷ C5×C11×C61 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 3355 |
A minimal faithful permutation representation — 2 generators on 77 points:
(2,3)(4,5)(7,11,12,10,15,8,9,16,13,14)(18,22,23,37,35,19,20,28,30,27)(21,24,34,41,46,25,26,32,50,60)(29,57,58,67,52,39,40,45,74,36)(31,64,65,62,63,43,44,66,42,61)(33,59,70,77,69,47,48,76,75,53)(38,73,68,54,71,55,56,72,51,49);(1,2,4,5,3)(6,7,10,15,9,12,16,11,14,13,8)(17,18,21,27,36,53,60,34,51,30,42,56,74,48,57,75,70,63,50,39,20,24,33,49,72,58,43,28,38,44,66,65,64,55,23,31,45,68,71,47,26,22,29,41,61,76,77,40,59,67,73,62,37,54,32,46,69,52,35,25,19)Polycyclic presentation:
2,5^2,11,61|c^a=c^4,d^a=d^10,e^a=e^60,d^b=d^4,e^b=e^20355 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33550, 60);