Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #6171⋊C3 |
| Fitting | C22×C27⋊C3 ⋊ C2 |
| Sylow | C27⋊C3 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D27⋊C3 |
| commutator, cosocle | C2×C27 | C22×C3 |
| Frattini | C2×C9 | C2×C3×D3 |
| Fitting | C22×C27⋊C3 | C2 |
| socle | C2×C3 | C2×C3×D9 |
| hypercenter | C22 | D27⋊C3 |
| hypocenter | C27 | C3×D4 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C22 | |
74 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3,2,3^3|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 | #88399⋊C3, D4×D27⋊C3, #88401⋊C3, … |
| maximal normal subgroup of | #88399⋊C3, D4×D27⋊C3, #88401⋊C3, … |
| maximal quotient of | ((C27⋊C3)⋊C4)⋊C4, (C2×D27⋊C3)⋊C4, #88368⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 157);