Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42×C7 · Schur multiplier (H2) = C22 |
| direct product | C3×C7×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C42 |
| commutator | C4 | C3×C42×C7 |
| Frattini, cosocle | C22×C4 | C22×C3×C7 |
| socle | C2×C3×C7 | #95 |
| prime radical | C23×C3×C7 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ G | |
172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,14,18,16,19,15,24)(13,20,22,17,23,25,21,26);(11,13,15,22,16,23,14,21)(12,17,18,20,19,26,24,25)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 778);