Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3×C7 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×C7×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C25×C3×C7 |
| socle | C23×C3×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4×C7 ◁ G | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2)(13,14,15)(16,17,18,19,20,21,22);(3,4);(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11680);