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) = C22 |
| direct product | C3×C7×#440 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C2×D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| prime radical | C2×C3×C7×D4 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C3×C42×C7 ◁ G | |
240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,14,13)(15,16)(17,19)(18,20)(21,22);(16,19)(17,22)(20,21);(11,13,14,12)(16,20)(19,21)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6240);