Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4179 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C9 ⋊ C3×C8 |
| Sylow | #732 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4179 |
| commutator, hypocenter | C3×C9 | C3×C8 |
| Frattini | C32×C4 | C3×D3 |
| Fitting | C4×#732 | C2 |
| socle | C2×C3 | (C3⋊Dic3)⋊C3 |
| prime radical | C2×#732 | C4 |
| cosocle | C3×C4×C9 | C2×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4 | |
49 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=fg^252 conjugacy classes — view the full character table.
| maximal subgroup of | #4185⋊C8, #4178⋊C8, #13046446, … |
| maximal normal subgroup of | #4185⋊C8, #4178⋊C8, #13046446, … |
| maximal quotient of | #13046322, C2×#10619505, #14515620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 24);