Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#2126 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #99 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×D4 | C23×C53 |
| socle | C2×C53 | C2×#99 |
| prime radical | C53×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
258 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1378 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 858);