Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3×C27 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×D27 |
| commutator, Fitting, cosocle | C2×C3×C27 | C22 |
| Frattini | C2×C9 | D32 |
| socle, prime radical | C2×C32 | C2×D9 |
| hypocenter | C3×C27 | Q8 |
| LCS | G ▷ C2×C3×C27 ▷ C3×C27 | |
| UCS | C1 ◁ C2 | |
57 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,d^3=f,f^3=g,b^a=bc,e^a=e^2,d^b=d^2f^2g^2,f^b=f^2g^2,g^b=g^287 conjugacy classes — view the full character table.
| maximal subgroup of | Dic6×D27, (C3⋊D27)⋊Q8, D3×Dic54, … |
| maximal normal subgroup of | Dic6×D27, (C3⋊D27)⋊Q8, D3×Dic54, … |
| maximal quotient of | Dic27⋊Dic3, #13046475, Dic3⋊Dic27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 58);