Finite Group DatabasebetaFinite Group Databasebeta| order | 20124 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#20269443 |
| derived | C3×C13×C43 ⋊ C3×C4 |
| Hall | C3×#13131976 ⋊ C4 |
| Hall | C13×C43 ⋊ C3×Dic3 |
| Sylow | C43 ⋊ C3⋊(Dic13⋊C3) |
| Sylow | C13 ⋊ C3⋊((C43⋊C3)⋊C4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C3⋊#16809817 |
| commutator, hypocenter | C3×C13×C43 | C3×C4 |
| Fitting, socle, cosocle | C2×C3×C13×C43 | C2×C3 |
| prime radical | C2×C3×#13131976 | C2 |
| LCS | G ▷ C3×C13×C43 | |
| UCS | C1 ◁ C2 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 6708 |
A minimal faithful permutation representation — 2 generators on 63 points:
(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(45,49,50,46,47,51)(48,55,54,52,53,56)(58,59)(60,61,62,63);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,45,48,47,51,54,53,55,56,50,49,52,46)(57,58,59)Polycyclic presentation:
2,3^2,13,43,2|a^2=f,c^a=c^2,d^a=d^12,e^a=e^42,d^b=d^3,e^b=e^36576 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20124, 97);