Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×D32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | D16 |
| commutator, Frattini, cosocle | C16 | C22×C3×C7 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ C3×C4×C7 ◁ C3×C7×C8 ◁ C3×C7×C16 ◁ G | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,36)(19,35)(20,34)(21,33)(22,32)(23,31)(24,30)(25,29)(26,28)399 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 811);