Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #560⋊ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #560⋊ΓL(1,𝔽8) |
| Fitting, Hall | C7×#560 ⋊ C3 |
| derived | C42×C7 ⋊ C22×C3 |
| Sylow | #560 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #560⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C7 | C22×C3 |
| Frattini | C22 | C22×C22⋊ΓL(1,𝔽8) |
| Fitting | C7×#560 | C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
134 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^2,3,2^2,7,2^2|d^2=g,e^2=gh,d^a=dg,e^a=egh,d^b=dgh,e^b=eh,d^c=de,e^c=dgh,f^c=f^4,g^c=h,h^c=gh55 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#560⋊C3, #72277370 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#560⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10074);