Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#89804 |
| Fitting | C3×C7×#120 ⋊ C2 |
| Hall | C3×C7 ⋊ B(4,𝔽2) |
| Sylow | C7 ⋊ C3×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)(4,5,7,9,10,8,6)(15,18);(11,13)(12,15)(14,17)(16,18);(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)174 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7×B(4,𝔽2), D3×#89804, #72244296, … |
| maximal normal subgroup of | C3×D7×B(4,𝔽2), D3×#89804, #72244296, … |
| maximal quotient of | #72246124, ΓL(1,𝔽8)×#89804, #72246190 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 4754);