Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C7×C8 · Schur multiplier (H2) = C4 |
| direct product | C3×C7×#322 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C42×C7 | C22 |
| commutator | C2 | C3×C4×C7×C8 |
| Frattini, cosocle | C42 | C22×C3×C7 |
| socle, prime radical | C22×C3×C7 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C42×C7 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,16,14,15,17,18);(12,15)(16,18)(19,20,21,23,22,24,25,26)840 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 762);