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×#444 |
| 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 | C4×D4 |
| prime radical | C3×C7×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ C2×C3×C7×C8 ◁ G | |
208 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)(11,12,15,19,16,13,20,18)(14,21,22,25,23,17,24,26);(11,13)(12,16)(14,21)(15,18)(17,23)(19,20)(22,25)(24,26);(11,14,15,22,16,23,20,24)(12,17,19,26,13,21,18,25)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6244);