Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
| direct product | C2×#16811515 |
| Sylow | C53 ⋊ C2×#347 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×#6157 |
| commutator | C4×C53 | C2×C42 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C2×C42×C53 | C4 |
| socle | C22×C53 | #97 |
| hypercenter | C2×C42 | #6157 |
| hypocenter | C53 | C2×#347 |
| LCS | G ▷ C4×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 | |
168 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
460 conjugacy classes — view the full character table.
| maximal subgroup of | #28298679, #28298681, #28298682, … |
| maximal normal subgroup of | #28298679, #28298681, #28298682, … |
| maximal quotient of | C2×#20289633, #28307787, #28307841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20726);