Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | #90410⋊C3 |
| Fitting | C7×#127 ⋊ C2×C3 |
| Hall | #90410 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #540 |
| Sylow | C7 ⋊ C3×#540 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#127 | C2×C3 |
| hypercenter | #127 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#540 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #127 | |
480 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 480 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^4,3,2^2,7|b^2=g,c^2=f,c^a=cg,d^a=df,h^a=h^6,d^b=dg,d^c=df,h^e=h^280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6742);