Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20269514⋊C2 |
| unambiguous semidirect product | #20269520⋊C2 |
| unambiguous semidirect product | #20269526⋊C2 |
| Fitting, socle, derived, Hall | C11×C61 ⋊ C22×C5 |
| Hall | #16809824 ⋊ 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:
(1,2,8,33,22)(3,13,16,48,27)(4,15,44,11,39)(5,21,26,58,28)(6,24,53,40,50)(7,9,36,49,41)(10,12,23,51,17)(14,45,18,46,52)(19,30,60,43,57)(20,55,32,31,54)(25,35,61,29,37)(38,47,56,42,59)(63,67,68,66,71,64,65,72,69,70);(2,3)(4,6)(5,9)(7,12)(8,16)(10,19)(11,14)(13,25)(15,20)(17,31)(18,26)(21,24)(22,28)(23,40)(27,44)(29,37)(30,46)(32,36)(33,49)(34,43)(35,38)(39,48)(41,57)(42,55)(45,50)(47,59)(51,58)(52,56)(53,61)(54,60)(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^964 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 39);