Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | D42×ΓL(1,𝔽8) |
| unambiguous semidirect product | (C7×D42)⋊C3 |
| Fitting, Hall | C7×D42 ⋊ C3 |
| Sylow | C7 ⋊ C3×D42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×D42 | C3 |
| hypercenter | D42 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×D42 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ D42 | |
856 conjugacy classes of subgroups; only the 273 normal subgroups are shown.
This group has 856 conjugacy classes of subgroups — too many to draw. Only its 273 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3,4)(10,11,12)(13,15,14);(2,4)(9,10,11,13,12,14,15);(5,6,7,8);(6,8)125 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9995);