Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (Q8⋊C9)⋊C9 |
| Sylow | Q8 ⋊ C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical | C2×C32 | C3×A4 |
| commutator | C3×Q8 | C3×C9 |
| Fitting | C3×Q8×C9 | C3 |
| cosocle | C32×Q8 | C32 |
| hypercenter | C2×C3×C9 | A4 |
| hypocenter | Q8 | C9⋊C9 |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C3×C9 | |
61 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
3^4,2^3|a^3=d,b^3=c,e^2=g,f^2=g,b^a=bd,e^a=f,f^a=efg,f^e=fg93 conjugacy classes — view the full character table.
| maximal subgroup of | #3688⋊C9, #3689⋊C9, C2×(Q8⋊C9)⋊C9, … |
| maximal normal subgroup of | #3688⋊C9, #3689⋊C9, C2×(Q8⋊C9)⋊C9, … |
| maximal quotient of | C2×(Q8⋊C9)⋊C9, #14515852, (Q8⋊C9)⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 91);