Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C72 · Schur multiplier (H2) = C7 |
| direct product | C72×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C72 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C72 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C72 ◁ C4×C72 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 784 |
A minimal faithful permutation representation — 2 generators on 30 points:
(1,2,3,4,5,6,7)(15,16,17,18,19,20,21,22)(23,30,29,28,27,26,25,24);(8,9,10,11,12,13,14)(15,23,19,27)(16,24,20,28)(17,25,21,29)(18,26,22,30)343 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 103);