Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×C53⋊#564 |
| unambiguous semidirect product | (C2×C53)⋊#564 |
| Sylow | C53 ⋊ C2×#564 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×#125×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×#125 | D53 |
| hypocenter | C53 | C2×#564 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×#125 | |
896 conjugacy classes of subgroups; only the 469 normal subgroups are shown.
This group has 896 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.
766 conjugacy classes — view the full character table.
| maximal subgroup of | #27276546, #27276547, C2×#20274016, … |
| maximal normal subgroup of | #27276546, #27276547, C2×#20274016, … |
| maximal quotient of | C2×#20277897, C2×#20277898, C2×#20277899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18653);