Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90168⋊C3 |
| Fitting | C2×C42×C7 ⋊ C2×C3 |
| Hall | #90168 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #532 |
| Sylow | C7 ⋊ C3×#532 |
620 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^4,3,2^2,7|b^2=g,d^2=f,c^a=cg,d^a=df,h^a=h^6,h^e=h^2116 conjugacy classes — view the full character table.
| maximal subgroup of | #72229384, #72229413, #72229420, … |
| maximal normal subgroup of | #72229384, #72229413, #72229420, … |
| maximal quotient of | #72229458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6500);