Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of (C22⋊C81)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (C22⋊C81)⋊C2 |
| derived | C22⋊C81 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊C81 | C2 |
| Frattini | C27 | S4 |
| Fitting | C22×C27 | D3 |
| socle | C22×C3 | D27 |
| LCS | G ▷ C22⋊C81 | |
| UCS | C1 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 162 |
Polycyclic presentation:
2,3^4,2^2|b^3=c,c^3=de,d^3=e^2,b^a=b^2c^2d^2,c^a=c^2d^2,d^a=d^2e,e^a=e^2,f^a=g,g^a=f,f^b=g,g^b=fg57 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 41);