Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | #126×#6157 |
| Fitting | #126×C53 ⋊ C4 |
| Sylow | C53 ⋊ C4×#126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#6157 |
| commutator, socle | C22×C53 | C23×C4 |
| Fitting | #126×C53 | C4 |
| hypercenter | #126 | #6157 |
| hypocenter | C53 | C4×#126 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ #126 | |
516 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
238 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#126×#6157, #529×#6157, #538×#6157, … |
| maximal normal subgroup of | C2×#126×#6157, #529×#6157, #538×#6157, … |
| maximal quotient of | #388×#6157, #387×#6157, #383×#6157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20978);