Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C53 ⋊ #1768 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C22×C53 | C2×C42 |
| Fitting | #95×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | #95 | #6157 |
| hypocenter | C53 | #1768 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ #95 | |
260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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=f,c^2=de,f^2=g,b^a=bd,c^a=cg,h^b=h^30,h^f=h^52316 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20290503, #28318260, C4×#16811629, … |
| maximal normal subgroup of | C2×#20290503, #28318260, C4×#16811629, … |
| maximal quotient of | #28302048, #28302051, #28302054, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20298);