Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #89260⋊C3 |
| Hall | #89260 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #330 |
| Sylow | C7 ⋊ C3×#330 |
120 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 40 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:
2^2,3,2^4,7|a^2=g,b^2=d,d^2=f,e^2=fg,b^a=be,d^a=dg,e^a=efg,h^a=h^6,e^b=ef,h^c=h^277 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 27);