Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C2×C16×C53 ⋊ C4 |
| Sylow | C53 ⋊ #2108 |
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×#6157 |
| commutator, socle | C2×C53 | C2×C4×C8 |
| Frattini | C8 | C22×#6157 |
| Fitting | C2×C16×C53 | C4 |
| hypercenter | C2×C16 | #6157 |
| hypocenter | C53 | #2108 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C16 ◁ C2×C16 | |
184 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|a^2=d,b^2=e,d^2=f,f^2=g,c^b=cg,h^b=h^30,h^e=h^52496 conjugacy classes — view the full character table.
| maximal subgroup of | #28299170, #28299173, #28299174, … |
| maximal normal subgroup of | #28299170, #28299173, #28299174, … |
| maximal quotient of | #28307755, #28307756, #28307768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20718);