Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C11⋊#11927414 |
| Fitting, socle, derived, Hall | C5×C11×C61 ⋊ C4 |
| Hall | C11×C61 ⋊ AGL(1,𝔽5) |
| Hall | C5×C61 ⋊ Dic11 |
| Sylow | C61 ⋊ C11⋊AGL(1,𝔽5) |
| Hall | C5×C11 ⋊ Dic61 |
| Sylow | C11 ⋊ #11927414 |
| Sylow | C5 ⋊ Dic671 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C11×C61 | C4 |
| prime radical, cosocle | D5×C11×C61 | C2 |
| LCS | G ▷ C5×C11×C61 | |
| UCS | C1 | |
24 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)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(63,64)(65,66)(67,68)(69,70)(71,72)(74,75,76,77);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(62,63,65,67,69,71,72,70,68,66,64)(73,74,75,77,76)Polycyclic presentation:
2^2,5,11,61|a^2=b,c^a=c^2,d^a=d^10,e^a=e^60,c^b=c^41345 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 86);