Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q82×D53 |
| Fitting | Q82×C53 ⋊ C2 |
| Sylow | C53 ⋊ C2×Q82 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D53 |
| commutator, socle | C22×C53 | C25 |
| Fitting | Q82×C53 | C2 |
| hypercenter | Q82 | D53 |
| hypocenter | C53 | C2×Q82 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
700 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#1623, D53×#1633, D53×#1645, … |
| maximal normal subgroup of | D53×#1623, D53×#1633, D53×#1645, … |
| maximal quotient of | Q8×C4⋊C4×D53, D53×#2369, Q8×#13133182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 18084);