Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | ΓL(1,𝔽8)×#537 |
| unambiguous semidirect product | (C7×#537)⋊C3 |
| Fitting, Hall | C7×#537 ⋊ C3 |
| Sylow | C7 ⋊ C3×#537 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#537 | C3 |
| hypercenter | #537 | ΓL(1,𝔽8) |
| hypocenter | C7 | C3×#537 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #537 | |
588 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(2,3,4)(5,7,6)(9,11)(12,13)(14,18)(15,16)(17,19);(1,2,3,5,4,6,7)(13,16)(17,18);(8,9)(10,11)(12,13)(14,17)(15,16)(18,19);(12,14,15,19)(13,17,16,18)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 9987);