Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4×C7 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×C7×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C7×C8 | C22 |
| commutator | C2 | C23×C3×C4×C7 |
| Frattini, cosocle | C4 | C24×C3×C7 |
| socle | C22×C3×C7 | C22×C4 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7×C8 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(19,20,21)(22,23,24,25,26,27,28);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18);(5,11)(8,14)(12,17)(15,18)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11276);