Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #2162 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×#6157 |
| commutator | C2×C53 | C22×C16 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C4×C8×C53 | C4 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C4×C8 | #6157 |
| hypocenter | C53 | #2162 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C4×C8 | |
160 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 78 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,c^2=d,e^2=f,f^2=g,b^a=bd,h^a=h^30,h^b=h^52,h^e=h^52496 conjugacy classes — view the full character table.
| maximal subgroup of | #28298269, #28298278, #28298279, … |
| maximal normal subgroup of | #28298269, #28298278, #28298279, … |
| maximal quotient of | #28310421, C4×#16811518, C4×#16811544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20869);