Finite Group DatabasebetaFinite Group Databasebeta| order | 33550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20269521 |
| 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 ⋊ D5×C61⋊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)(18,19,21,25,33)(20,23,29,41,38)(22,27,37,54,47)(24,31,30,43,62)(26,35,50,34,48)(28,39,57,73,56)(32,45,58,36,52)(40,59,69,60,74)(42,61,63,49,53)(44,64,75,66,72)(46,65,76,71,77)(51,68,55,70,67);(1,2,4,5,3)(6,7,10,15,9,12,16,11,14,13,8)(17,18,20,24,32,46,54,29,42,25,34,49,66,39,58,73,57,48,41,45,19,22,28,40,60,75,70,43,63,35,51,50,67,59,21,26,36,53,61,65,23,30,44,37,55,71,77,64,76,52,69,68,62,74,27,38,56,72,33,47,31)Polycyclic presentation:
2,5^2,11,61|c^a=c^4,d^a=d^10,d^b=d^4,e^b=e^20373 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33550, 66);