Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C24 |
| direct product | C3×C7×#556 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C2×C3×C7×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12)(13,15)(14,17)(16,18)(21,25)(23,26);(11,13,16,14)(12,17,18,15)(19,20,22,24)(21,23,25,26);(11,14,16,13)(12,17,18,15)(19,21)(20,23)(22,25)(24,26);(11,15,16,17)(12,13,18,14)525 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11265);