Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #90303⋊C3 |
| Fitting | C7×#121 ⋊ C2×C3 |
| Hall | #90303 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #546 |
| Sylow | C7 ⋊ C3×#546 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#121 | C2×C3 |
| hypercenter | #121 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#546 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #121 | |
668 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|b^2=g,c^2=g,b^a=bg,h^a=h^6,d^b=dg,d^c=df,h^e=h^289 conjugacy classes — view the full character table.
| maximal subgroup of | #72229315, AGL(1,𝔽7)×#546, #72230275, … |
| maximal normal subgroup of | #72229315, AGL(1,𝔽7)×#546, #72230275, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6637);