Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89913⋊C3 |
| Fitting | C7×#131 ⋊ C2×C3 |
| Hall | #89913 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #471 |
| Sylow | C7 ⋊ C3×#471 |
208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^3,3,2^3,7|b^2=f,e^2=g,f^2=g,b^a=bf,c^a=ce,e^a=eg,f^a=fg,h^a=h^6,e^c=eg,h^d=h^268 conjugacy classes — view the full character table.
| maximal subgroup of | #72229611, #72229686, #72229707 |
| maximal normal subgroup of | #72229611, #72229686, #72229707 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1330);