Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×D53⋊(C22⋊C8) |
| unambiguous semidirect product | D53⋊(C2×C22⋊C8) |
| Sylow | C53 ⋊ C22×C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×#6157 |
| commutator | C2×C53 | C23×C8 |
| Frattini | C22 | C23×#6157 |
| Fitting | C23×C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C23×C4 | #6157 |
| hypocenter | C53 | C22×C22⋊C8 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 | |
1072 conjugacy classes of subgroups; only the 306 normal subgroups are shown.
This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 306 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
496 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20289552, #28300315, #28300319, … |
| maximal normal subgroup of | C2×#20289552, #28300315, #28300319, … |
| maximal quotient of | C2×D53⋊#323, C2×D53⋊#324, D53⋊#1458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21084);