Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10619770 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, derived | C22×C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C22×C3×C9 | C2×C3 |
| Frattini | C32 | D3×A4 |
| socle | C22×C32 | C3×D3 |
| LCS | G ▷ C22×C3×C9 | |
| UCS | C1 | |
108 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3,2^2,3^3|e^3=g,e^a=e^2g^2,f^a=f^2,g^a=g^2,c^b=d,d^b=cd,e^b=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊S4)⋊C9)⋊C2, C2×#10619770, A4×#4169, … |
| maximal normal subgroup of | ((C3⋊S4)⋊C9)⋊C2, C2×#10619770, A4×#4169, … |
| maximal quotient of | #13047043, #13047044, #13047056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 289);