Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #89926⋊C3 |
| Fitting | C2×C7×D8 ⋊ C2×C3 |
| Hall | #89926 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #496 |
| Sylow | C7 ⋊ C3×#496 |
288 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 72 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:
2^3,3,2^3,7|a^2=f,e^2=g,b^a=bf,c^a=cfg,h^a=h^6,c^b=ce,e^b=eg,e^c=eg,h^d=h^262 conjugacy classes — view the full character table.
| maximal subgroup of | #72229408, #72229686, AGL(1,𝔽7)×#496, … |
| maximal normal subgroup of | #72229408, #72229686, AGL(1,𝔽7)×#496, … |
| maximal quotient of | #72229429, #72229547, #72259499, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1340);