Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#4174 |
| Fitting | C4×AΣL(1,𝔽27) ⋊ C2 |
| Sylow | AΣL(1,𝔽27) ⋊ D4 |
134 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3,2,3^3|b^2=d,b^a=bd,e^a=e^2,f^a=f^2g,e^c=ef,f^e=fg61 conjugacy classes — view the full character table.
| maximal subgroup of | #13046413, #13046414, #13046417, … |
| maximal normal subgroup of | #13046413, #13046414, #13046417, … |
| maximal quotient of | #13046555, #13046558, #13046559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 129);