Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Sylow | C32×C9 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×C3⋊D3 |
| commutator, Fitting, cosocle | C2×C32×C9 | C22 |
| Frattini | C2×C3 | D3×C3⋊D3 |
| socle, prime radical | C2×C33 | C2×D3 |
| hypocenter | C32×C9 | Q8 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,f^3=g,b^a=bc,f^a=f^2g^2,g^a=g^2,d^b=d^2,e^b=e^269 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6189, #13048292, #13048306, … |
| maximal normal subgroup of | D3×#6189, #13048292, #13048306, … |
| maximal quotient of | Dic9⋊(C3⋊Dic3), #13047706, #13047710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 386);