Finite Group DatabasebetaFinite Group Databasebeta| order | 33550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20269522 |
| Fitting, socle, derived | C5×C11×C61 ⋊ C2×C5 |
| Hall | C5×#16809823 ⋊ 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)(20,23,29,38,55)(22,27,37,53,46)(24,31,42,58,28)(26,35,49,34,47)(30,40,57,70,62)(32,44,48,63,39)(36,51,68,56,64)(41,52,69,74,73)(43,59,60,67,75)(45,61,72,77,76)(50,66,54,71,65);(1,2,4,5,3)(6,7,10,15,9,12,16,11,14,13,8)(17,18,20,24,32,45,27,38,56,21,26,36,52,70,63,40,57,47,29,39,33,46,62,75,59,69,66,42,51,49,65,35,50,67,25,34,48,64,68,76,55,58,73,37,54,72,61,74,77,44,60,71,31,43,53,23,30,41,19,22,28)Polycyclic presentation:
2,5^2,11,61|c^a=c^4,d^a=d^10,d^b=d^4,e^b=e^58373 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33550, 65);