Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C7×H(5,𝔽2) ⋊ C2×C3 |
| Hall | #90016 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ B(4,𝔽2) |
| Sylow | C7 ⋊ C3×B(4,𝔽2) |
396 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
Polycyclic presentation:
2^3,3,2^3,7|b^a=bf,c^a=ce,h^a=h^6,e^b=eg,f^c=fg,h^d=h^265 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×B(4,𝔽2), #72214654, #72215138, … |
| maximal normal subgroup of | AGL(1,𝔽7)×B(4,𝔽2), C2×#13052144, C3×#13052144, … |
| maximal quotient of | #72215138, C2×#13052144, #72246127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 1433);