Finite Group DatabasebetaFinite Group Databasebeta| order | 12642 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(D43⋊C7) |
| unambiguous semidirect product | (ΓL(1,𝔽8)⋊D43)⋊C7 |
| Fitting, socle, derived | C7×C43 ⋊ C2×C3×C7 |
| Hall | ΓL(1,𝔽8)×C43⋊C7 ⋊ C2 |
| Hall | C7⋊(D43⋊C7) ⋊ C3 |
| Hall | C7×C43⋊C7 ⋊ C2×C3 |
| Sylow | C43 ⋊ C7×AGL(1,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C43 | C2×C3×C7 |
| LCS | G ▷ C7×C43 | |
| UCS | C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 301 |
A minimal faithful permutation representation — 2 generators on 50 points:
(2,5,6,11,14,30,35,3,4,7,9,19,22,42)(8,18,21,41,10,26,27,12,13,29,33,16,17,40)(15,38,39,43,23,28,32,24,25,31,34,36,20,37)(45,48,49,46,47,50);(1,2,5,8,6,10,18,24,11,20,26,22,21,34,25,39,14,23,37,33,27,35,42,40,41,32,36,19,31,38,43,29,30,17,28,9,15,13,16,7,12,4,3)(44,45,47,50,49,48,46)Polycyclic presentation:
2,3,7^2,43|d^a=d^6,e^a=e^42,d^b=d^2,e^c=e^4164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12642, 37);