Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#1996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×C22⋊C4 |
| commutator | C23 | C22×C4×C53 |
| Frattini, cosocle | C24 | C23×C53 |
| prime radical | C24×C53 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
324 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 728);