Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of D49 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C49⋊C16 |
| derived, Sylow | C49 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D49 |
| commutator, hypocenter | C49 | C16 |
| Frattini | C7×C8 | D7 |
| Fitting, cosocle | C8×C49 | C2 |
| socle, prime radical | C2×C7 | C7⋊C8 |
| LCS | G ▷ C49 | |
| UCS | C1 ◁ C8 | |
15 conjugacy classes of subgroups.
| minimal degree | μ(G) = 65 |
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^4,7^2|a^2=b,b^2=c,c^2=d,e^7=f,e^a=e^6f^6,f^a=f^6208 conjugacy classes — view the full character table.
| maximal subgroup of | C16×D49, #13095838, C2×C49⋊C16, … |
| maximal normal subgroup of | C16×D49, #13095838, C2×C49⋊C16, … |
| maximal quotient of | C49⋊C32, C2×C49⋊C16 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 1);