Finite Group DatabasebetaFinite Group Databasebeta| order | 13420 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C11⋊#11927413 |
| 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 ⋊ #6923 |
| Sylow | C11 ⋊ #11927413 |
| Sylow | C5 ⋊ C11⋊#6923 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C11×C61 | C4 |
| prime radical, cosocle | C11×D305 | 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:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,49)(36,51,52,60)(40,55,56,58)(44,59,57,54)(46,61,53,50)(63,64)(65,66)(67,68)(69,70)(71,72)(74,75,76,77);(1,2,5,10,19,32,49,61,42,25,12,3,7,14,23,36,53,45,60,59,38,21,8,16,27,40,57,43,28,41,58,55,34,17,29,44,56,39,26,15,24,37,54,51,30,46,52,35,22,13,6,11,20,33,50,47,48,31,18,9,4)(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^11,c^b=c^4,e^b=e^60850 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13420, 85);