Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×#13133157 |
| Sylow | C53 ⋊ C4×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C2×D53 |
| commutator | C2×C53 | C2×C4×C8 |
| Frattini | C2×C8 | C22×D53 |
| Fitting | C2×C4×C8×C53 | C2 |
| socle | C22×C53 | C22×C8 |
| hypercenter | C2×C4×C8 | D53 |
| hypocenter | C53 | C4×M32 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4×C8 ◁ C2×C4×C8 | |
196 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 149 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 | #27270923, #27270924, #27270925, … |
| maximal normal subgroup of | #27270923, #27270924, #27270925, … |
| maximal quotient of | C8×#13133157, #27293085, #27293114, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6104);