Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C4⋊#4185 |
| Fitting | C4×#732 ⋊ C2 |
| Sylow | #732 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4185 |
| commutator, cosocle | C2×#732 | C22 |
| Frattini | C2×C32 | C2×C3⋊D3 |
| Fitting | C4×#732 | C2 |
| socle | C2×C3 | C2×#272 |
| hypercenter | C4 | #4185 |
| hypocenter | #732 | D4 |
| LCS | G ▷ C2×#732 ▷ #732 | |
| UCS | C1 ◁ C2 ◁ C4 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^4|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,g^a=g^2,e^d=ef,f^d=fg,f^e=fg^243 conjugacy classes — view the full character table.
| maximal subgroup of | #13046449, #13046451, #13046452, … |
| maximal normal subgroup of | #13046449, #13046451, #13046452, … |
| maximal quotient of | #13046753, #13046756, #13046757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 187);