Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C53 ⋊ #2183 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C22×C8 |
| Fitting | C4⋊C8×C53 | C4 |
| hypercenter | C4⋊C8 | #6157 |
| hypocenter | C53 | #2183 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4⋊C8 | |
126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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=d,c^2=de,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^23,h^b=h^52,h^c=h^30,h^e=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | D53⋊#2183, C2×#20291132, #28328427, … |
| maximal normal subgroup of | D53⋊#2183, C2×#20291132, #28328427, … |
| maximal quotient of | #28310780, #28310784, #28310786, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20927);