Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #1905 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16810499 |
| commutator | C2×C4×C53 | C22×C4 |
| Frattini | C22×C4 | C22×D53 |
| Fitting | C53×#421 | C2 |
| socle | C2×C53 | #386 |
| hypercenter | #421 | D53 |
| hypocenter | C53 | #1905 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #421 | |
310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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=f,b^2=df,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^b=eg,d^c=dg,e^c=eg,f^c=fg,h^c=h^52601 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1905, C2×#20275206, #27385986, … |
| maximal normal subgroup of | D53×#1905, C2×#20275206, #27385986, … |
| maximal quotient of | #27279507, #27279521, #27279535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5001);