Finite Group DatabasebetaFinite Group Databasebeta| order | 784 |
| isoclinism | isoclinic class of C7⋊D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | C7⋊(C7⋊C16) |
| derived, Sylow | C72 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | C7⋊D7 |
| commutator, hypocenter | C72 | C16 |
| Fitting, cosocle | C72×C8 | C2 |
| socle, prime radical | C2×C72 | C8 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C8 | |
50 conjugacy classes of subgroups.
| transitive degree | μt(G) = 784 |
Polycyclic presentation:
2^4,7^2|a^2=b,b^2=c,c^2=d,e^a=e^6,f^a=f^6208 conjugacy classes — view the full character table.
| maximal subgroup of | #13096032, D7×C7⋊C16, #13096035, … |
| maximal normal subgroup of | #13096032, D7×C7⋊C16, #13096035, … |
| maximal quotient of | C7⋊(C7⋊C32), C2×C7⋊(C7⋊C16) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(784, 42);