Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | D53⋊(C22⋊C16) |
| Sylow | C53 ⋊ C2×C22⋊C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×#6157 |
| commutator | C2×C53 | C22×C16 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C22×C8×C53 | C4 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C22×C8 | #6157 |
| hypocenter | C53 | C2×C22⋊C16 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 | |
272 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 94 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,e^2=f,f^2=g,b^a=bd,h^a=h^30,h^c=h^52,h^e=h^52496 conjugacy classes — view the full character table.
| maximal subgroup of | #28298405, #28298414, #28298452, … |
| maximal normal subgroup of | #28298405, #28298414, #28298452, … |
| maximal quotient of | D53⋊#1314, D53⋊#1322, #28308004, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20735);