Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | D53⋊#338 |
| Sylow | C53 ⋊ C2×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C4×#6157 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C2×M16×C53 | C4 |
| socle | C2×C53 | C2×#95 |
| hypercenter | C2×M16 | #6157 |
| hypocenter | C53 | C2×#338 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×M16 | |
372 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 128 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=e,b^2=f,d^2=g,e^2=g,f^2=g,b^a=bd,f^a=fg,h^a=h^30,e^b=eg,h^c=h^52,h^e=h^52304 conjugacy classes — view the full character table.
| maximal subgroup of | #28298646, #28298654, #338×#6157, … |
| maximal normal subgroup of | #28298646, #28298654, #338×#6157, … |
| maximal quotient of | #28300110, #28300112, #28300114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20184);