Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C3×C7×#421 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C2×D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle | C2×C3×C7 | C2×C22⋊C4 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ C22×C3×C4×C7 ◁ G | |
308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,15,18,16,19,14,23)(13,20,22,17,21,26,24,25);(11,13)(12,17)(14,24)(15,22)(16,21)(18,26)(19,25)(20,23);(13,21)(17,25)(20,26)(22,24)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6221);