Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#16810526 |
| Sylow | C53 ⋊ C2×#390 |
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D53 |
| commutator | C22×C53 | C23×C4 |
| Frattini | C23 | C23×D53 |
| Fitting | C23×D4×C53 | C2 |
| socle | C24×C53 | C23 |
| hypercenter | C23×D4 | D53 |
| hypocenter | C53 | C2×#390 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C24 ◁ C23×D4 | |
1360 conjugacy classes of subgroups; only the 405 normal subgroups are shown.
This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 405 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 | #27289455, #27289457, #27290356, … |
| maximal normal subgroup of | #27289455, #27289457, #27290356, … |
| maximal quotient of | C2×#20272944, C2×#20272952, C4×#16810526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6967);