Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | #131×Dic53 |
| Sylow | C53 ⋊ C4×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D53 |
| commutator | C2×C53 | C22×C42 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C2×#131×C53 | C2 |
| socle | C22×C53 | C23×C4 |
| hypercenter | C2×#131 | D53 |
| hypocenter | C53 | C4×#131 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#131 | |
572 conjugacy classes of subgroups; only the 377 normal subgroups are shown.
This group has 572 conjugacy classes of subgroups — too many to draw. Only its 377 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1120 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #350×Dic53, #27299147, #27300027, … |
| maximal normal subgroup of | #350×Dic53, #27299147, #27300027, … |
| maximal quotient of | #131×#88322, C8×#13133224, #27349702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 12931);