Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#89849 |
| Fitting | C3×C7×#127 ⋊ C2 |
| Hall | C3×C7 ⋊ #453 |
| Sylow | C7 ⋊ C3×#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)(4,5,7,9,10,8,6)(12,15)(13,18)(16,17);(13,18)(15,16);(5,6)(7,8)(9,10)(11,12)(13,15)(14,17)(16,18)174 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×#453, D3×#89849, #72244311, … |
| maximal normal subgroup of | C3×D7×#453, D3×#89849, #72244311, … |
| maximal quotient of | ΓL(1,𝔽8)×#89849, #72246297, C3×#13056714 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4799);