Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20269517⋊C2 |
| unambiguous semidirect product | #20269523⋊C2 |
| unambiguous semidirect product | #20269529⋊C2 |
| Fitting, socle, derived, Hall | C11×C61 ⋊ C22×C5 |
| Hall | #16809821 ⋊ 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:
(1,2,8,29,35)(3,12,41,59,43)(4,9,18,20,34)(5,19,24,55,25)(6,21,52,54,47)(7,26,45,22,30)(10,33,38,60,17)(11,36,14,15,46)(13,42,16,39,51)(23,56,61,32,49)(27,58,31,44,28)(37,53,50,48,57)(63,67,68,66,71,64,65,72,69,70);(2,3)(4,6)(5,9)(7,11)(8,14)(10,17)(12,22)(13,20)(15,27)(16,30)(18,31)(19,34)(21,37)(23,40)(24,29)(25,35)(26,32)(28,45)(33,39)(36,49)(38,50)(41,54)(42,47)(43,46)(44,56)(48,53)(51,59)(52,61)(55,60)(57,58)(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^3464 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 40);