Finite Group DatabasebetaFinite Group Databasebeta| order | 7776 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 124416 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3×C4 |
| abelian | C3×C4×C8×C81 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C4×C27 | C22×C32 |
| socle, prime radical | C22×C32 | C2×C4×C27 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
396 conjugacy classes of subgroups.
7776 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C3×C4×C16×C81, C3×C82×C81, C2×C3×C4×C8×C81, … |
| maximal normal subgroup of | C3×C4×C16×C81, C3×C82×C81, C2×C3×C4×C8×C81, … |
| maximal quotient of | C3×C4×C16×C81, C3×C82×C81, C2×C3×C4×C8×C81, … |
Past the SmallGroups library coverage — no SmallGroups id.