Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C7⋊(#543⋊C3) |
| unambiguous semidirect product | #543⋊ΓL(1,𝔽8) |
| Fitting, derived, Hall | C7×#543 ⋊ C3 |
| Sylow | #543 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #543⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, Fitting, cosocle, hypocenter | C7×#543 | C3 |
| socle | C22×C7 | C22⋊A4 |
| LCS | G ▷ C7×#543 | |
| UCS | C1 ◁ C22 | |
144 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
3,2^4,7,2^2|d^2=g,e^2=g,b^a=c,c^a=bc,d^a=e,e^a=de,f^a=f^4,e^b=eh,d^c=dh,e^d=eg62 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#543⋊C3, #72277085 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#543⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10093);