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) = C24 |
| direct product | C3×C7×#541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C23×C3×C7 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2,3)(4,5,6,7,8,9,10)(13,17)(15,18)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(11,12,14,16)(13,15,17,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(11,13)(12,15)(14,17)(16,18)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11250);