Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#4181 |
| Fitting | C4×C32⋊C9 ⋊ C2 |
| Sylow | C32⋊C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×C3⋊D9 |
| commutator, cosocle | C2×C32⋊C9 | C22 |
| Frattini, socle | C2×C32 | C2×C3⋊D3 |
| Fitting | C4×C32⋊C9 | C2 |
| hypercenter | C3×C4 | C3⋊D9 |
| hypocenter | C32⋊C9 | D4 |
| LCS | G ▷ C2×C32⋊C9 ▷ C32⋊C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
158 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^2,f^a=f^2,e^d=eg75 conjugacy classes — view the full character table.
| maximal subgroup of | #13046377, #13046379, #13046381, … |
| maximal normal subgroup of | #13046377, #13046379, #13046381, … |
| maximal quotient of | #13046733, #13046736, #13046737, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 175);