Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C22⋊C4×#88324 |
| Sylow | C53 ⋊ C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#6157 |
| commutator | C2×C53 | C2×C4×C8 |
| Fitting | C2×C22⋊C4×C53 | C4 |
| socle | C23×C53 | C22×C4 |
| hypercenter | C2×C22⋊C4 | #6157 |
| hypocenter | C53 | C8×C22⋊C4 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
384 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
340 conjugacy classes — view the full character table.
| maximal subgroup of | #99×#88324, #28301574, #28301586, … |
| maximal normal subgroup of | #99×#88324, #28301574, #28301586, … |
| maximal quotient of | C22⋊C8×#88324, #28301497, #28301511, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 20243);