Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C3×C7×#413 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle | C2×C3×C7 | C2×C22⋊C4 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ C22×C3×C4×C7 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(11,13)(12,17)(14,22)(15,24)(16,23)(18,25)(19,21)(20,26);(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6213);