Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4177 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C9 ⋊ C3×C8 |
| Sylow | #731 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4177 |
| commutator, hypocenter | C3×C9 | C3×C8 |
| Frattini | C32×C4 | C3×D3 |
| Fitting | C4×#731 | C2 |
| socle | C2×C3 | (C3⋊Dic3)⋊C3 |
| prime radical | C2×H(3,𝔽3) | Dic3 |
| cosocle | C3×C4×C9 | C2×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4 | |
46 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3,2^2,3^3|a^2=c,c^2=d,e^3=g,e^a=e^2g^2,f^a=f^2,g^a=g^2,e^b=ef,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #4184⋊C8, #4176⋊C8, #13046428, … |
| maximal normal subgroup of | #4184⋊C8, #4176⋊C8, #13046428, … |
| maximal quotient of | #13046320, C2×#10619503, #14515621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 22);