Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C22×C42 |
| direct product | C3×C7×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,15,18)(13,20,21,17)(14,22,24,26)(16,19,25,23);(11,13,15,21)(12,17,18,20)(14,23,24,19)(16,22,25,26);(11,14)(12,13)(15,24)(16,17)(18,21)(19,22)(20,25)(23,26);(11,13)(12,14)(15,21)(16,22)(17,19)(18,24)(20,23)(25,26)399 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11270);