Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C72 · Schur multiplier (H2) = C23×C7 |
| direct product | C2×C72×D4 |
| unambiguous semidirect product | C4⋊(C22×C72) |
| unambiguous semidirect product | C22⋊(C22×C72) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C72 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C72 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C72 ◁ G | |
270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 392 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(7,8,9,10,11,12,13);(3,4,5,6)(14,15,16,17,18,19,20);(4,6)490 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 157);