Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C53 ⋊ #2029 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133185 |
| commutator | C24×C53 | C23 |
| Frattini | C24 | C22×D53 |
| Fitting | C53×#379 | C2 |
| socle | C22×C53 | #120 |
| hypercenter | #379 | D53 |
| hypocenter | C53 | #2029 |
| LCS | G ▷ C24×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #379 | |
466 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 466 conjugacy classes of subgroups — too many to draw. Only its 69 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=d,b^a=bd,c^a=ce,f^a=fg,c^b=cf,e^b=eg,h^b=h^52754 conjugacy classes — view the full character table.
| maximal subgroup of | #27271670, #27271672, #27296809, … |
| maximal normal subgroup of | #27271670, #27271672, #27296809, … |
| maximal quotient of | #27287060, #27287062, #27287065, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 5674);