Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C49 · Schur multiplier (H2) = C22 |
| direct product | C2×Q8×C49 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C49 | C22 |
| commutator | C2 | C23×C49 |
| Frattini, cosocle | C2×C7 | C23×C7 |
| socle, prime radical | C22×C7 | C22×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C49 ◁ G | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 3 generators on 59 points:
(1,2)(11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 39);