Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C3×#90281 |
| Fitting | C3×C7×#120 ⋊ C2 |
| Hall | C3×C7 ⋊ D42 |
| Sylow | C7 ⋊ C3×D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D7 |
| commutator, cosocle | C22×C7 | C24×C3 |
| Frattini | C22 | C23×C3×D7 |
| Fitting | C3×C7×#120 | C2 |
| socle | C22×C3×C7 | C24 |
| hypercenter | C3×#120 | D7 |
| hypocenter | C7 | C3×D42 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22×C3 ◁ C3×#120 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2,3)(4,5,7,9,10,8,6)(12,14)(15,16)(17,18);(11,12)(13,14)(16,18);(12,14);(5,6)(7,8)(9,10)(16,18)201 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 10222);