Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | C2×ΓL(1,𝔽8)×#121 |
| unambiguous semidirect product | (C2×C7×#121)⋊C3 |
| Fitting, Hall | C2×C7×#121 ⋊ C3 |
| Sylow | C7 ⋊ C2×C3×#121 |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(12,13,14)(15,17,16);(4,6)(8,10)(11,12,13,15,14,16,17);(7,8,9,10);(3,4)(5,6)140 conjugacy classes — view the full character table.
| maximal subgroup of | #72259836, C2×#121×AGL(1,𝔽7) |
| maximal normal subgroup of | #72259836, C2×#121×AGL(1,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9972);