Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #127⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #384⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#384 ⋊ C3 |
| derived | C42×C7 ⋊ C3×C4 |
| Sylow | #384 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #384⋊C3 |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2,3,2^3,7,2^2|a^2=c,d^2=g,e^2=gh,d^a=dg,e^a=egh,d^b=de,e^b=dgh,f^b=f^2,g^b=h,h^b=gh76 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4354);