Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20269516⋊C2 |
| unambiguous semidirect product | #20269522⋊C2 |
| unambiguous semidirect product | #20269528⋊C2 |
| Fitting, socle, derived, Hall | C11×C61 ⋊ C22×C5 |
| Hall | #16809823 ⋊ C22 |
| Hall | D11×D61 ⋊ C5 |
| Sylow | C61 ⋊ C2×AGL(1,𝔽11) |
| Sylow | C11 ⋊ C2×D61⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C11×C61 | C22×C5 |
| LCS | G ▷ C11×C61 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 671 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,8,28,31)(3,11,38,12,40)(4,13,45,17,48)(5,16,49,55,22)(6,18,53,61,47)(7,23,15,33,51)(9,29,21,57,27)(10,32,60,14,46)(19,54,59,35,58)(20,52,36,41,56)(24,25,43,50,34)(26,44,37,30,42)(63,67,68,66,71,64,65,72,69,70);(2,3)(4,6)(5,8)(7,10)(9,14)(11,19)(12,17)(13,24)(15,27)(16,21)(18,33)(20,36)(22,31)(23,41)(25,43)(26,38)(28,35)(29,48)(30,39)(32,50)(34,52)(37,53)(40,47)(42,46)(44,45)(49,56)(51,59)(54,61)(55,57)(58,60)(62,63,66,71,65,68,72,67,70,69,64)Polycyclic presentation:
2^2,5,11,61|e^a=e^60,d^b=d^10,d^c=d^4,e^c=e^5864 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 41);