Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90436⋊C3 |
| Fitting | C2×C7×M16 ⋊ C2×C3 |
| Hall | #90436 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #568 |
| Sylow | C7 ⋊ C3×#568 |
516 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|d^2=f,f^2=g,c^a=cg,h^a=h^6,d^b=dg,h^e=h^2122 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#568, #72229363, #72229450, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#568, #72229363, #72229450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6768);