Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D53⋊C64 |
| derived, Sylow | C53 ⋊ C2×C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #6157 |
| commutator, hypocenter | C53 | C2×C64 |
| Frattini | C16 | C2×#6157 |
| Fitting | C32×C53 | C4 |
| socle | C2×C53 | C2×C32 |
| LCS | G ▷ C53 | |
| UCS | C1 ◁ C32 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^30,h^b=h^52,h^c=h^52544 conjugacy classes — view the full character table.
| maximal subgroup of | C64×#6157, #28299311, #28299376, … |
| maximal normal subgroup of | C64×#6157, #28299311, #28299376, … |
| maximal quotient of | #28299310, #28299319, #28299385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19678);