Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C28 |
| direct product | C22×#13133124 |
| Sylow | C53 ⋊ C22×#95 |
| subgroup | G/H | |
|---|---|---|
| center | C25 | C2×D53 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C23 | C23×D53 |
| Fitting | C24×C4×C53 | C2 |
| socle | C25×C53 | C22 |
| hypercenter | C24×C4 | D53 |
| hypocenter | C53 | C22×#95 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C25 ◁ C24×C4 | |
1880 conjugacy classes of subgroups; only the 981 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 981 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 | #27272011, #27272012, #27272013, … |
| maximal normal subgroup of | #27272011, #27272012, #27272013, … |
| maximal quotient of | C2×C4×#13133124, C2×#20272937, C2×#20272938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6834);