Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of C3⋊D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊(C27⋊C8) |
| unambiguous semidirect product | C27⋊(C3⋊C8) |
| derived, Sylow | C3×C27 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊D27 |
| commutator, hypocenter | C3×C27 | C8 |
| Frattini | C4×C9 | C3⋊D3 |
| Fitting, cosocle | C3×C4×C27 | C2 |
| socle, prime radical | C2×C32 | Dic9 |
| LCS | G ▷ C3×C27 | |
| UCS | C1 ◁ C4 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^3=f,f^3=g,d^a=d^2f^2g^2,e^a=e^2,f^a=f^2g^2,g^a=g^2168 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×D27, #13046392, D3×C27⋊C8, … |
| maximal normal subgroup of | C3⋊C8×D27, #13046392, D3×C27⋊C8, … |
| maximal quotient of | C3⋊(C27⋊C16), C2×C3⋊(C27⋊C8), C9⋊(C27⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 27);