Finite Group DatabasebetaFinite Group Databasebeta| order | 33550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20269529 |
| Fitting, socle, derived | C5×C11×C61 ⋊ C2×C5 |
| Hall | C5×#16809821 ⋊ 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)(18,22,23,36,41,19,20,27,29,51)(21,33,34,42,62,25,26,47,48,59)(24,43,44,45,58,30,31,61,40,63)(28,56,57,76,35,38,39,70,71,50)(32,53,54,75,65,46,37,66,67,52)(49,69,68,72,73,55,64,77,60,74);(1,2,4,5,3)(6,7,10,15,9,12,16,11,14,13,8)(17,18,21,27,37,55,47,26,35,41,58,71,75,64,54,68,73,44,62,65,29,42,60,70,56,46,30,22,28,40,43,31,45,38,20,24,32,39,57,72,48,36,52,59,61,74,77,66,69,67,76,63,51,50,33,34,49,53,23,25,19)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^34355 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33550, 62);