Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D81 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic81 |
| unambiguous semidirect product | (C2×C81)⋊C4 |
| derived, Sylow | C81 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D81 |
| commutator, hypocenter | C81 | C2×C4 |
| Frattini | C2×C27 | C2×D3 |
| Fitting | C22×C81 | C2 |
| socle, prime radical | C22×C3 | D27 |
| cosocle | C2×C81 | C22 |
| LCS | G ▷ C81 | |
| UCS | C1 ◁ C22 | |
40 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic81, Dic81⋊C4, C4⋊Dic81, … |
| maximal normal subgroup of | C4×Dic81, Dic81⋊C4, C4⋊Dic81, … |
| maximal quotient of | C2×C81⋊C8, #13046292, C4×Dic81, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 7);