Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C27⋊C3)⋊Q8 |
| unambiguous semidirect product | Dic54⋊C3 |
| Sylow | C27⋊C3 ⋊ Q8 |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3,2,3^3|a^2=d,b^2=d,e^3=f,f^3=g,b^a=bd,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2,e^c=eg75 conjugacy classes — view the full character table.
| maximal subgroup of | Dic108⋊C3, #88360⋊C3, #88368⋊C3, … |
| maximal normal subgroup of | Dic108⋊C3, #88360⋊C3, #88368⋊C3, … |
| maximal quotient of | ((C27⋊C3)⋊C4)⋊C4, C4⋊((C27⋊C3)⋊C4), C2×(C27⋊C3)⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 123);