Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90459⋊C3 |
| Fitting | C7×#135 ⋊ C2×C3 |
| Hall | #90459 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #575 |
| Sylow | C7 ⋊ C3×#575 |
524 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|f^2=g,d^a=dg,h^a=h^6,c^b=cg,d^b=dg,d^c=df,f^c=fg,f^d=fg,h^e=h^280 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#575, #72229627, #72229639, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#575, #72229627, #72229639, … |
| maximal quotient of | C4×#6474⋊C3, C4×#6477⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6791);