Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32⋊C9 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ((C3⋊D3)⋊C9)⋊C2 |
| commutator, Fitting, cosocle | C2×C32⋊C9 | C22 |
| Frattini, socle | C2×C32 | D32 |
| prime radical | C2×C33 | C2×D3 |
| hypocenter | C32⋊C9 | Q8 |
| LCS | G ▷ C2×C32⋊C9 ▷ C32⋊C9 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,d^3=f,b^a=bc,e^a=e^2,g^a=g^2,d^b=d^2f^2,f^b=f^2,g^b=g^2,e^d=eg39 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊Q8, #4181⋊Q8, #4169⋊Q8, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊Q8, #4181⋊Q8, #4169⋊Q8, … |
| maximal quotient of | #13046470, #13046471, #13046472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 57);