Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C8 |
| direct product | C53×#1720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C53 |
| prime radical | #127×C53 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
186 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 72 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 | C2×C53×#1720, C53×#19613, C53×#19619, … |
| maximal normal subgroup of | C2×C53×#1720, C53×#19613, C53×#19619, … |
| maximal quotient of | C53×#8821, C53×#9827, C53×#9828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 452);