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) = C22 |
| direct product | C3×C4×C7×Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle, prime radical | C22×C3×C7 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ C3×C42×C7 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(4,5,6,7)(24,25,26,27,28,29,30);(8,9,10,11,12,13,14,15)(16,23,22,21,20,19,18,17);(8,16,12,20)(9,17,13,21)(10,18,14,22)(11,19,15,23)588 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6239);