Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊(C3⋊C8))⋊C9 |
| derived | C32 ⋊ C8×C9 |
| Sylow | C32⋊C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C8×C9 |
| Frattini | C32×C4 | C3×D3 |
| Fitting | C4×C32⋊C9 | C2 |
| socle | C2×C32 | C3×Dic3 |
| prime radical | C2×C33 | C3×C4 |
| cosocle | C33×C4 | C2×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C3×C4 | |
62 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3,2,3^2|a^2=c,b^3=d,c^2=e,f^a=f^2,g^a=g^2,f^b=fg120 conjugacy classes — view the full character table.
| maximal subgroup of | #4181⋊C8, #4169⋊C8, #13046374, … |
| maximal normal subgroup of | #4181⋊C8, #4169⋊C8, #13046374, … |
| maximal quotient of | (C3⋊(C3⋊C16))⋊C9, C2×(C3⋊(C3⋊C8))⋊C9, #14515618, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 13);