Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C4×#13133186 |
| Fitting | C24×C4×C53 ⋊ C2 |
| Sylow | C53 ⋊ C4×#120 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D53 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C23 | C23×D53 |
| Fitting | C24×C4×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C24×C4 | D53 |
| hypocenter | C53 | C4×#120 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 | |
1224 conjugacy classes of subgroups; only the 277 normal subgroups are shown.
This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 277 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 | #27278774, #27278784, #27389584, … |
| maximal normal subgroup of | #27278774, #27278784, #27389584, … |
| maximal quotient of | C4×#16810502, C4×#16810496, C4×#16810479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7012);