Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of C22⋊(C7⋊(C42⋊C3)) |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C23×C4 |
| unambiguous semidirect product | C22⋊(C7⋊(C42⋊C3)) |
| unambiguous semidirect product | C7⋊(C22⋊(C42⋊C3)) |
| Fitting, derived, Hall | C22×C42×C7 ⋊ C3 |
| Sylow | C22×C42 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C22⋊(C42⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C42×C7 | C3 |
| Frattini | C22 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| socle | C24×C7 | A4 |
| LCS | G ▷ C22×C42×C7 | |
| UCS | C1 | |
206 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
3,2^4,7,2^2|d^2=h,e^2=gh,b^a=c,c^a=bc,d^a=e,e^a=deg,f^a=f^4,g^a=h,h^a=gh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10092);