Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | M32×C53 ⋊ C4 |
| Sylow | C53 ⋊ C2×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×C4×#6157 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C8 | C22×#6157 |
| Fitting | M32×C53 | C4 |
| socle | C2×C53 | C2×#97 |
| hypercenter | M32 | #6157 |
| hypocenter | C53 | C2×#347 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ M32 | |
168 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 80 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=e,d^2=f,f^2=g,b^a=bf,e^a=eg,d^b=dg,h^b=h^30,h^c=h^52,h^e=h^52304 conjugacy classes — view the full character table.
| maximal subgroup of | #347×#6157, #28310201, #28310507, … |
| maximal normal subgroup of | #347×#6157, #28310201, #28310507, … |
| maximal quotient of | #28307779, #28307783, #28307784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20725);