Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #99 |
| commutator | C2×C4 | C42×C53 |
| Frattini, cosocle | C22×Q8 | C22×C53 |
| prime radical | C23×C53 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C53 ◁ C22×Q8×C53 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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.
| maximal subgroup of | C53×#7404, C53×#7405, C53×#7409, … |
| maximal normal subgroup of | C53×#7404, C53×#7405, C53×#7409, … |
| maximal quotient of | C53×#7189, C53×#7206, C53×#7215, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 126);