Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20269515⋊C2 |
| unambiguous semidirect product | #20269521⋊C2 |
| unambiguous semidirect product | #20269527⋊C2 |
| Fitting, socle, derived, Hall | C11×C61 ⋊ C22×C5 |
| Hall | #16809822 ⋊ 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:
(1,2,8,30,11)(3,13,43,61,44)(4,15,50,21,20)(5,6,22,58,19)(7,26,34,53,33)(9,17,41,52,14)(10,35,31,51,37)(12,38,56,16,45)(18,23,59,36,25)(24,27,39,46,60)(28,48,49,40,55)(29,57,42,32,47)(63,67,68,66,71,64,65,72,69,70);(2,3)(4,6)(5,9)(7,12)(8,10)(11,19)(13,23)(14,21)(15,27)(16,18)(17,31)(20,36)(22,39)(24,41)(25,34)(26,46)(28,48)(29,42)(30,44)(32,43)(33,55)(35,38)(37,53)(40,57)(45,47)(49,50)(51,58)(52,56)(54,60)(59,61)(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^2064 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 42);