Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10620206 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, socle, Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3⋊(C3⋊(C3⋊D3)) | C22 |
| Fitting, socle, hypocenter | C34 | D4 |
| LCS | G ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34 | |
| UCS | C1 | |
277 conjugacy classes of subgroups.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^3,3^4|b^a=bc,d^a=d^2,g^a=g^2,d^b=e,e^b=d,f^b=g,g^b=f,d^c=d^2,e^c=e^2,f^c=f^2,g^c=g^227 conjugacy classes — view the full character table.
| maximal subgroup of | #13049791, #13049803, #13049804, … |
| maximal normal subgroup of | #13049791, #13049803, #13049804, … |
| maximal quotient of | #13049455, #13049456, #13049457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 725);