Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#1683 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C53 |
| prime radical | C2×D4×C53 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C42×C53 ◁ G | |
160 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
1537 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C53×#1683, C53×#19110, C53×#19126, … |
| maximal normal subgroup of | C2×C53×#1683, C53×#19110, C53×#19126, … |
| maximal quotient of | C53×#8648, C53×#9215, C53×#9219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 415);