Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×C7×#104 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle | C22×C3×C7 | C22⋊C4 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ C22×C3×C4×C7 ◁ G | |
340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(11,12,13)(14,15,16,17,18,19,20);(3,4,6,9,7,5,10,8);(3,5)(4,7)(6,8)(9,10)588 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6220);