Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #90348⋊C3 |
| Fitting | C7×#123 ⋊ C2×C3 |
| Hall | #90348 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #538 |
| Sylow | C7 ⋊ C3×#538 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#123 | C2×C3 |
| hypercenter | #123 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#538 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #123 | |
500 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|c^2=f,d^a=dg,h^a=h^6,c^b=cg,d^b=df,h^e=h^280 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#538, #72276683, #72276718, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#538, #72276683, #72276718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6683);