Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#1490 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C42 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C4 |
| prime radical | C22×D4×C53 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G | |
316 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
2014 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C53×#8367, C53×#8370, C53×#8372, … |
| maximal normal subgroup of | C53×#8367, C53×#8370, C53×#8372, … |
| maximal quotient of | C53×#7606, C53×#7607, C53×#7608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 222);