Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D27⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C27⋊C3)⋊C8 |
| unambiguous semidirect product | (C27⋊C8)⋊C3 |
| derived | C27 ⋊ C3×C8 |
| Sylow | C27⋊C3 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D27⋊C3 |
| commutator, hypocenter | C27 | C3×C8 |
| Frattini | C4×C9 | C3×D3 |
| Fitting | C4×C27⋊C3 | C2 |
| socle | C2×C3 | C3×Dic9 |
| prime radical | C2×C32 | Dic9 |
| cosocle | C4×C27 | C2×C3 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C4 | |
38 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=f,f^3=g,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2,e^b=eg84 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D27⋊C3, #88359⋊C3, C2×(C27⋊C3)⋊C8, … |
| maximal normal subgroup of | C8×D27⋊C3, #88359⋊C3, C2×(C27⋊C3)⋊C8, … |
| maximal quotient of | (C27⋊C3)⋊C16, C2×(C27⋊C3)⋊C8, #6864⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 18);