Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#5198 |
| Fitting | C3×C7×AGL(1,ℤ/8) ⋊ C2 |
| Hall | C3×C7 ⋊ #453 |
| Sylow | C3 ⋊ C7×#453 |
336 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,3,4,5,6,7)(8,9,10)(12,15)(13,18)(16,17);(11,12)(13,15)(14,16)(17,18);(9,10)(13,18)(15,17)189 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C7×#453, D7×#5198, #72244338, … |
| maximal normal subgroup of | D3×C7×#453, D7×#5198, #72244338, … |
| maximal quotient of | C7×#10591522 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5524);