Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C24 |
| direct product | C7×#4741 |
| Fitting | C3×C7×#99 ⋊ C2 |
| Hall | C3×C7 ⋊ B(4,𝔽2) |
| Sylow | C3 ⋊ C7×B(4,𝔽2) |
396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 396 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)(11,12)(13,17,18,15)(14,16);(11,13)(12,15)(14,18)(16,17);(9,10)(11,12)(13,15)(14,16)(17,18)189 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#4741, D3×C7×B(4,𝔽2), #72244304, … |
| maximal normal subgroup of | D7×#4741, D3×C7×B(4,𝔽2), #72244304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 5065);