Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C2×C42 |
| direct product | C4×#13133271 |
| Fitting | C4×C8×C53 ⋊ C4 |
| Sylow | C53 ⋊ C4×#97 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×#6157 |
| commutator | C2×C53 | C43 |
| Frattini | C2×C4 | C22×#6157 |
| Fitting | C4×C8×C53 | C4 |
| socle | C22×C53 | C2×C42 |
| hypercenter | C4×C8 | #6157 |
| hypocenter | C53 | C4×#97 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C42 ◁ C4×C8 | |
360 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
496 conjugacy classes — view the full character table.
| maximal subgroup of | #28298292, C8×#13133271, #28300035, … |
| maximal normal subgroup of | #28298292, C8×#13133271, #28300035, … |
| maximal quotient of | C4×#16811464, C4×#88324⋊C8, C8×#13133271, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20164);