Finite Group DatabasebetaFinite Group Databasebeta| order | 13140 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C3⋊#13091732) |
| Fitting, socle, derived, Hall | C32×C5×C73 ⋊ C4 |
| Hall | C32×C73 ⋊ AGL(1,𝔽5) |
| Hall | C5×C73 ⋊ C3⋊Dic3 |
| Sylow | C73 ⋊ C3⋊ΓL(1,𝔽16) |
| Hall | C32×C5 ⋊ #64408 |
| Sylow | C32 ⋊ #13091732 |
| Sylow | C5 ⋊ C3⋊(C3⋊#64408) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C32×C5×C73 | C4 |
| prime radical, cosocle | C32×D365 | C2 |
| LCS | G ▷ C32×C5×C73 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 84 |
| transitive degree | μt(G) = 3285 |
A minimal faithful permutation representation — 3 generators on 84 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)(32,47,48,60)(36,51,52,61)(40,49,55,57)(44,58,59,54)(46,62,63,50)(53,64,65,69)(56,66,67,70)(68,71,72,73)(75,76,77,78)(80,81)(83,84);(1,2,5,10,19,32,49,34,17,29,44,60,69,73,72,67,55,39,26,15,24,37,54,62,42,25,12,3,7,14,23,36,53,66,51,30,46,63,45,61,70,65,52,35,22,13,6,11,20,33,50,58,38,21,8,16,27,40,56,68,71,64,47,59,43,28,41,57,48,31,18,9,4)(74,75,76,78,77)(82,83,84);(79,80,81)Polycyclic presentation:
2^2,3^2,5,73|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^27,e^b=e^4,f^b=f^72831 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13140, 139);