Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#13133206 |
| Fitting | C23×Q8×C53 ⋊ C2 |
| Sylow | C53 ⋊ C22×#124 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C23×Q8×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C23×Q8 | D53 |
| hypocenter | C53 | C22×#124 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×Q8 | |
1880 conjugacy classes of subgroups; only the 645 normal subgroups are shown.
This group has 1880 conjugacy classes of subgroups — too many to draw. Only its 645 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1096 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27276727, #27290633, #27290636, … |
| maximal normal subgroup of | #27276727, #27290633, #27290636, … |
| maximal quotient of | C22×#16810488, C22×#16810513, C22×#16810524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17759);