Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C49 · Schur multiplier (H2) = C1 |
| direct product | Q16×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C49 | D4 |
| commutator | C4 | C22×C49 |
| Frattini, cosocle | C4×C7 | C22×C7 |
| socle, prime radical | C2×C7 | C7×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C49 ◁ C4×C49 ◁ G | |
27 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 2 generators on 65 points:
(1,2,3,4,5,6,7,8)(9,16,15,14,13,12,11,10)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65);(1,9,5,13)(2,10,6,14)(3,11,7,15)(4,12,8,16)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 26);