Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C7 · Schur multiplier (H2) = C25 |
| direct product | C3×C7×#575 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C24×C3×C7 |
| socle | C2×C3×C7 | C22×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ C22×C3×C4×C7 ◁ G | |
524 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)(13,20)(14,19)(15,23)(16,18)(17,21)(22,26)(24,25);(11,13)(12,17)(14,21)(15,24)(16,22)(18,25)(19,26)(20,23);(12,18)(17,25)(19,23)(20,26);(11,14,16,15)(12,19,18,23)(13,21,22,24)(17,26,25,20)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11284);