Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #89905⋊C3 |
| Fitting | C2×C7×M16 ⋊ C2×C3 |
| Hall | #89905 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #417 |
| Sylow | C7 ⋊ C3×#417 |
324 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 104 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:
2^3,3,2^3,7|b^2=g,c^2=e,e^2=g,f^2=g,c^a=cf,e^a=eg,f^a=fg,h^a=h^6,c^b=cg,h^d=h^286 conjugacy classes — view the full character table.
| maximal subgroup of | #72218332, C4×#6471⋊C3, #72218598, … |
| maximal normal subgroup of | C4×#6471⋊C3, #72218598, AGL(1,𝔽7)×#417, … |
| maximal quotient of | #72218332 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1321);