Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C32⋊C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | C3⋊D9 |
| commutator, hypocenter | C32⋊C9 | C8 |
| Frattini | C32×C4 | C3⋊D3 |
| Fitting, cosocle | C4×C32⋊C9 | C2 |
| socle | C2×C32 | C3⋊Dic3 |
| prime radical | C2×C33 | Dic3 |
| LCS | G ▷ C32⋊C9 | |
| UCS | C1 ◁ C3×C4 | |
82 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^3=f,d^a=d^2f^2,e^a=e^2,f^a=f^2,e^d=eg84 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C9)⋊C8, #4169⋊C8, #13046375, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C9)⋊C8, #4169⋊C8, #13046375, … |
| maximal quotient of | #13046324, C2×#10619507, #14515657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 26);