Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C22⋊(C7⋊C16))⋊C3 |
| Hall | C22⋊(C7⋊C16) ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C22⋊C16 |
| Sylow | C7 ⋊ C3×C22⋊C16 |
132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 60 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:
2^2,3,2^4,7|a^2=e,e^2=f,f^2=g,b^a=bd,h^a=h^6,h^c=h^2152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 110);