Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#127×D53 |
| Fitting | C2×#127×C53 ⋊ C2 |
| Sylow | C53 ⋊ C22×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#127×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#127 | D53 |
| hypocenter | C53 | C22×#127 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
3000 conjugacy classes of subgroups; only the 661 normal subgroups are shown.
This group has 3000 conjugacy classes of subgroups — too many to draw. Only its 661 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
784 conjugacy classes — view the full character table.
| maximal subgroup of | #27275574, #27299576, C2×#331×D53, … |
| maximal normal subgroup of | #27275574, #27299576, C2×#331×D53, … |
| maximal quotient of | C2×#384×D53, C2×#390×D53, C4×#127×D53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17780);