Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C22×C22⋊#88322 |
| Sylow | C53 ⋊ C22×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D53 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C24×C4×C53 | C2 |
| socle | C24×C53 | C2×C4 |
| hypercenter | C24×C4 | D53 |
| hypocenter | C53 | C22×C22⋊C8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C4 | |
1072 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 469 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1744 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#20272557, C2×#20272558, #27273685, … |
| maximal normal subgroup of | C2×#20272557, C2×#20272558, #27273685, … |
| maximal quotient of | C22×#16810347, C22×#16810349, C2×#20273029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 12942);