Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C24 |
| direct product | C3×C7×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 1344 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2,3)(4,5,6,7,8,9,10)(13,19)(14,22)(16,20)(17,24)(27,28,31,34)(29,32,37,35)(30,38,39,33)(36,41,42,40);(11,12,15,18)(13,20,19,16)(14,17,22,24)(21,26,25,23)(27,28)(29,35)(30,38)(31,34)(32,37)(33,39)(36,40)(41,42);(11,13,15,19)(12,16,18,20)(14,21,22,25)(17,23,24,26)(27,29)(28,32)(30,36)(31,37)(33,40)(34,35)(38,41)(39,42);(11,14)(12,17)(13,21)(15,22)(16,23)(18,24)(19,25)(20,26)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)399 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11273);