Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42×C7 · Schur multiplier (H2) = C2×C42 |
| direct product | C3×C7×#376 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4×C7 | C22 |
| commutator | C2 | C2×C3×C42×C7 |
| Frattini, cosocle | C23 | C23×C3×C7 |
| socle, prime radical | C23×C3×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4×C7 ◁ G | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14)(19,20)(21,24)(22,23)(25,26);(15,16,17,18)(20,23)(24,26);(19,21,22,25)(20,24,23,26)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6176);