Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #399⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#399 ⋊ C3 |
| derived | C7×#95 ⋊ C2×C3 |
| Sylow | #399 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #399⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #127⋊ΓL(1,𝔽8) |
| commutator, cosocle, hypocenter | C7×#95 | C2×C3 |
| Frattini | C23 | C2×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#399 | C3 |
| socle | C23×C7 | C2×A4 |
| LCS | G ▷ C7×#95 | |
| UCS | C1 ◁ C2 | |
102 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2,3,2^2,7,2^3|c^2=f,d^2=fh,c^a=cg,d^a=dgh,c^b=cd,d^b=cgh,e^b=e^2,f^b=fgh,g^b=h,h^b=gh,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4360);