Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C3×C7×#424 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4×C7 | C23 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| socle | C22×C3×C7 | C22×C4 |
| prime radical | C23×C3×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4×C7 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3)(4,5,6,7,8,9,10)(12,14)(15,16,18,21,19,22,25,28)(17,20,24,27,23,26,29,30);(11,12,13,14)(15,17)(16,20)(18,24)(19,23)(21,27)(22,26)(25,29)(28,30);(17,23)(20,26)(24,29)(27,30)588 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6224);