Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C25 |
| direct product | C3×C7×#577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C24×C3×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C3×C7×#41 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,15,19)(13,17,20,24)(14,16,22,18)(21,23,26,25);(12,16)(14,22)(17,23)(18,19)(21,26)(24,25);(11,13)(12,17)(14,21)(15,20)(16,23)(18,25)(19,24)(22,26);(13,20)(17,24)(21,26)(23,25)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11286);