Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90439⋊C3 |
| Fitting | C2×C7×M16 ⋊ C2×C3 |
| Hall | #90439 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #575 |
| Sylow | C7 ⋊ C3×#575 |
524 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|c^2=g,d^2=f,f^2=g,d^a=df,f^a=fg,h^a=h^6,d^b=dg,h^e=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#575, #72229363, #72229461, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#575, #72229363, #72229461, … |
| maximal quotient of | C4×#6471⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6771);