Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #89268⋊C3 |
| Hall | #89268 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #354 |
| Sylow | C7 ⋊ C3×#354 |
140 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2^2,3,2^4,7|a^2=g,b^2=d,e^2=g,b^a=be,d^a=df,e^a=eg,h^a=h^6,e^b=efg,f^b=fg,h^c=h^2,e^d=eg50 conjugacy classes — view the full character table.
| maximal subgroup of | #72276750, AGL(1,𝔽7)×#354, #72276899 |
| maximal normal subgroup of | #72276750, AGL(1,𝔽7)×#354, #72276899 |
| maximal quotient of | #72277053 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 34);