Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C7⋊(Q8⋊SL(2,𝔽3)) |
| unambiguous semidirect product | Q8⋊(C7⋊SL(2,𝔽3)) |
| Fitting, derived, Hall | C7×Q82 ⋊ C3 |
| Sylow | Q82 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ Q8⋊SL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, Fitting, cosocle, hypocenter | C7×Q82 | C3 |
| socle | C22×C7 | C22⋊A4 |
| LCS | G ▷ C7×Q82 | |
| UCS | C1 ◁ C22 | |
124 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
3,2^4,7,2^2|b^2=h,c^2=h,d^2=g,e^2=g,b^a=c,c^a=bc,d^a=e,e^a=de,f^a=f^4,c^b=ch,e^d=eg69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10094);