Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊#4184 |
| Fitting | C4×#731 ⋊ C2 |
| Sylow | #731 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4184 |
| commutator, cosocle | C2×#731 | C22 |
| Frattini | C2×C32 | C2×C3⋊D3 |
| Fitting | C4×#731 | C2 |
| socle | C2×C3 | C2×#272 |
| hypercenter | C4 | #4184 |
| hypocenter | #731 | D4 |
| LCS | G ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C2 ◁ C4 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^4|b^2=c,e^3=g,b^a=bc,d^a=d^2,e^a=e^2g^2,f^a=fg^2,g^a=g^2,e^d=ef,f^d=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | #13046431, #13046433, #13046434, … |
| maximal normal subgroup of | #13046431, #13046433, #13046434, … |
| maximal quotient of | #13046748, #13046751, #13046752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 184);