Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C53 ⋊ #2150 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C22×#6157 |
| commutator, socle | C22×C53 | C22×C8 |
| Fitting | C4×C8×C53 | C4 |
| hypercenter | C4×C8 | #6157 |
| hypocenter | C53 | #2150 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C4×C8 | |
172 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 82 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,b^2=d,e^2=f,f^2=g,b^a=bd,c^a=cg,h^a=h^30,h^c=h^52,h^e=h^52472 conjugacy classes — view the full character table.
| maximal subgroup of | #28298302, #28298308, #28298354, … |
| maximal normal subgroup of | #28298302, #28298308, #28298354, … |
| maximal quotient of | #28309824, #28309850, #28309879, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20843);