Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#560⋊C3 |
| unambiguous semidirect product | #560⋊(C3×C7) |
| unambiguous semidirect product | (#126⋊C3)⋊(C2×C7) |
| Fitting, Hall | C7×#560 ⋊ C3 |
| derived | C42 ⋊ C22×C3×C7 |
| Sylow | #560 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | #560⋊C3 |
| commutator, cosocle, hypocenter | C42 | C22×C3×C7 |
| Frattini | C22 | C22×C7×A4 |
| Fitting | C7×#560 | C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C7 | |
134 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 32 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 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(9,19,20,14,10,22)(11,12,15)(13,23,16,17,18,21);(8,22)(9,18)(10,14)(11,16)(12,21)(13,19)(15,20)(17,23)119 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#560⋊C3, C7×#11710731 |
| maximal normal subgroup of | ΓL(1,𝔽8)×#560⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10045);