Finite Group DatabasebetaFinite Group Databasebeta| order | 33550 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C5⋊#20269523 |
| 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 ⋊ 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)(20,23,29,40,53)(22,27,37,52,47)(24,31,43,36,51)(26,35,50,34,48)(28,32,45,61,69)(30,42,39,56,64)(38,54,70,68,65)(41,57,62,49,66)(44,60,58,73,76)(46,63,75,72,77)(55,71,67,74,59);(1,2,4,5,3)(6,7,10,15,9,12,16,11,14,13,8)(17,18,20,24,32,46,50,23,30,33,47,64,76,57,45,62,66,37,53,69,25,34,49,67,71,60,70,51,42,52,68,27,38,55,19,22,28,39,56,72,40,31,44,48,65,77,75,73,63,61,74,54,43,59,35,29,41,58,21,26,36)Polycyclic presentation:
2,5^2,11,61|c^a=c^4,d^a=d^10,d^b=d^4,e^b=e^34373 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(33550, 64);