Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#16810525 |
| Fitting | C22×C42×C53 ⋊ C2 |
| Sylow | C53 ⋊ C2×#390 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C23 | C23×D53 |
| Fitting | C22×C42×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C22×C42 | D53 |
| hypocenter | C53 | C2×#390 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C22×C42 | |
1360 conjugacy classes of subgroups; only the 381 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 381 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1720 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#20272535, #27272076, #27272078, … |
| maximal normal subgroup of | C2×#20272535, #27272076, #27272078, … |
| maximal quotient of | C2×#20272945, C2×#20272953, C4×#16810525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6966);