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) = C22×C4 |
| direct product | C3×C4×C7×C4⋊C4 |
| unambiguous semidirect product | C4⋊(C3×C42×C7) |
| 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 | |
388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6,7)(16,17,18,19,20,21,22);(9,11)(12,13,14,15);(8,9,10,11)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6178);