Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1842 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C22×C53 | C22×C8 |
| Fitting | C2×C42×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C42 | #6157 |
| hypocenter | C53 | #1842 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
300 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 138 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=d,c^2=e,f^2=g,b^a=bd,c^a=ce,h^a=h^30,h^b=h^52,h^f=h^52472 conjugacy classes — view the full character table.
| maximal subgroup of | #28298041, #28298063, #28298084, … |
| maximal normal subgroup of | #28298041, #28298063, #28298084, … |
| maximal quotient of | #28303578, #28303579, #28303581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20443);