Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #393⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#393 ⋊ C3 |
| derived | C22×C7×Q8 ⋊ C2×C3 |
| Sylow | #393 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #393⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #120⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C22×C7×Q8 | C2×C3 |
| Frattini | C23 | C2×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#393 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C22×C7×Q8 | |
| UCS | C1 ◁ C2 | |
114 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2,3,2^4,7,2|c^2=h,d^2=h,c^a=ce,d^a=def,c^b=cd,d^b=c,e^b=f,f^b=ef,g^b=g^2,d^c=dh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4356);