Finite Group DatabasebetaFinite Group Databasebeta| order | 13140 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#13091733⋊C3) |
| unambiguous semidirect product | (C3⋊#13091733)⋊C3 |
| Fitting, socle, derived | C3×C5×C73 ⋊ C3×C4 |
| Hall | C3×C5×C73⋊C3 ⋊ C4 |
| Hall | C3×C73⋊C3 ⋊ AGL(1,𝔽5) |
| Hall | C5×C73 ⋊ C3×Dic3 |
| Sylow | C73 ⋊ C3×ΓL(1,𝔽16) |
| Sylow | C5 ⋊ C3⋊(#64408⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C5×C73 | C3×C4 |
| prime radical | C3×C5⋊(D73⋊C3) | C2 |
| cosocle | C3×D365 | C2×C3 |
| LCS | G ▷ C3×C5×C73 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1095 |
A minimal faithful permutation representation — 2 generators on 81 points:
(2,7,12,8,4,16,5,17,10,3,11,25)(6,27,43,28,14,23,15,46,30,13,31,33)(9,39,55,19,21,61,22,37,40,20,41,62)(18,42,70,57,35,65,36,66,59,34,60,47)(24,56,26,64,32,67,45,53,49,44,54,48)(29,68,72,38,51,71,52,63,58,50,69,73)(75,76,77,78)(80,81);(1,2,6,7,18,37,27,45,4,12,26,21,42,63,51,40,14,32,43,13,29,53,48,65,16,33,22,8,19,38,64,73,35,61,72,57,70,59,34,58,62,60,71,44,50,20,3,9,23,25,47,67,56,52,28,30,54,31,55,69,68,66,41,49,10,11,24,46,39,36,17,15,5)(74,75,76,78,77)(79,80,81)Polycyclic presentation:
2,3,2,3,5,73|a^2=c,d^a=d^2,e^a=e^2,f^a=f^46,f^b=f^8,e^c=e^4,f^c=f^72117 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13140, 98);