Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #4178⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4178⋊C4 |
| derived, Sylow | #732 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4178⋊C2 |
| commutator, hypocenter | #732 | C2×C4 |
| Frattini | C2×C32 | D32 |
| Fitting, cosocle | C2×#732 | C22 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| prime radical | C2×#4178 | C2 |
| LCS | G ▷ #732 | |
| UCS | C1 ◁ C2 | |
89 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^4|b^2=c,d^3=g,e^a=e^2,f^a=f^2g,d^b=d^2g^2,f^b=f^2,g^b=g^2,e^d=ef,f^e=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊Q8, #13047139, #13047141, … |
| maximal normal subgroup of | #4178⋊Q8, #13047139, #13047141, … |
| maximal quotient of | #4178⋊C8, #13046447, #13046494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 79);