Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#1506 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C4 | C23×C53 |
| prime radical | C23×C53 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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.
| maximal subgroup of | C53×#8413, C53×#8415, C53×#8492, … |
| maximal normal subgroup of | C53×#8413, C53×#8415, C53×#8492, … |
| maximal quotient of | C53×#7581, C53×#7588, C53×#7590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 238);