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×#563 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3×C7 |
| prime radical | C2×C3×C7×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ G | |
392 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)(12,15)(13,17)(19,20,24,22)(21,27,29,25)(23,30,32,26)(28,33,31,34);(12,15)(16,18)(19,21,24,29)(20,25,22,27)(23,31,32,28)(26,34,30,33);(11,12)(13,16)(14,15)(17,18)(19,22,24,20)(21,25,29,27)(23,30,32,26)(28,33,31,34);(11,13)(12,16)(14,17)(15,18)(19,23)(20,26)(21,28)(22,30)(24,32)(25,33)(27,34)(29,31)399 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11272);