Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C9⋊#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊#622 |
| derived, Sylow | C32×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#157 |
| commutator, hypocenter | C32×C9 | C8 |
| Frattini | C2×C3 | C3⋊#157 |
| Fitting | C2×C32×C9 | C4 |
| socle, prime radical | C2×C33 | Dic3 |
| cosocle | C9×C3⋊Dic3 | C2 |
| LCS | G ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
58 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,f^3=g,d^a=e,e^a=d^2,f^a=f^2g^2,g^a=g^2,d^b=d^2,e^b=e^260 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#622, #13048223, #13048224, … |
| maximal normal subgroup of | D9×#622, #13048223, #13048224, … |
| maximal quotient of | C9⋊#3708, C2×C9⋊#622, C9⋊#6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 353);