Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#16811653 |
| Sylow | C53 ⋊ C2×#435 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×#6157 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C22 | C23×#6157 |
| Fitting | C2×C4⋊C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C4⋊C4 | #6157 |
| hypocenter | C53 | C2×#435 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
564 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
316 conjugacy classes — view the full character table.
| maximal subgroup of | #28301587, #28318516, C4×#16811653, … |
| maximal normal subgroup of | #28301587, #28318516, C4×#16811653, … |
| maximal quotient of | C2×#20290315, C2×#20290323, C2×#20290319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21216);