Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4⋊Q8×Dic53 |
| Sylow | C53 ⋊ C4×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D53 |
| commutator | C22×C53 | C23×C4 |
| Fitting | C2×C4⋊Q8×C53 | C2 |
| socle | C23×C53 | C24 |
| hypercenter | C2×C4⋊Q8 | D53 |
| hypocenter | C53 | C4×C4⋊Q8 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
588 conjugacy classes of subgroups; only the 289 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 289 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
784 conjugacy classes — view the full character table.
| maximal subgroup of | #333×Dic53, #332×Dic53, #27275051, … |
| maximal normal subgroup of | #333×Dic53, #332×Dic53, #27275051, … |
| maximal quotient of | #389×Dic53, #391×Dic53, #384×Dic53, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 7083);