Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #89421⋊C3 |
| Hall | #89421 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C4×C4⋊C4 |
| Sylow | C7 ⋊ C3×C4×C4⋊C4 |
388 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^3,3,2^3,7|a^2=g,b^2=f,c^2=e,c^a=cg,h^a=h^6,c^b=cg,h^d=h^2140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 839);