Finite Group DatabasebetaFinite Group Databasebeta| order | 13140 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#13091732⋊C3) |
| unambiguous semidirect product | (C3⋊#13091732)⋊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:
(2,7,12,8,4,17,5,16,10,3,11,24)(6,26,48,27,14,44,15,38,28,13,29,34)(9,35,59,18,20,50,21,33,36,19,37,57)(22,61,72,47,40,71,41,64,62,39,63,73)(23,51,25,65,30,66,43,67,46,42,49,45)(31,68,55,69,53,32,54,58,70,52,60,56)(75,76,77,78)(80,81);(1,2,6,16,31,20,38,43,4,12,25,33,58,40,64,57,14,30,48,27,47,51,66,56,24,44,19,3,9,22,42,62,53,59,71,52,55,70,69,73,36,60,72,65,41,21,8,18,34,17,32,45,67,39,13,28,49,29,50,61,63,68,35,46,10,11,23,26,37,54,7,15,5)(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^27,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, 97);