Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C25 |
| direct product | C3×C7×#536 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C2×C3×C7×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3)(4,5,6,7,8,9,10)(12,14)(16,20)(17,23)(18,26)(28,30);(15,16,19,20)(17,24,23,21)(18,22,26,27)(25,30,29,28);(11,12)(13,14)(15,17,19,23)(16,21,20,24)(18,25,26,29)(22,28,27,30);(15,18)(16,22)(17,25)(19,26)(20,27)(21,28)(23,29)(24,30)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11245);