Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | ΓL(1,𝔽8)×#444 |
| unambiguous semidirect product | (C7×#444)⋊C3 |
| Fitting, Hall | C7×#444 ⋊ C3 |
| Sylow | C7 ⋊ C3×#444 |
208 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 99 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 — 3 generators on 23 points:
(2,3,4)(5,7,6)(8,9,12,16,13,10,17,15)(11,18,19,22,20,14,21,23);(1,2,3,5,4,6,7)(8,10)(9,13)(11,18)(12,15)(14,20)(16,17)(19,22)(21,23);(8,11,12,19,13,20,17,21)(9,14,16,23,10,18,15,22)110 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×AGL(1,ℤ/16), #72228676, AGL(1,𝔽7)×#444, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×AGL(1,ℤ/16), #72228676, AGL(1,𝔽7)×#444, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4264);