Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #66359⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #66359⋊C2 |
| derived | #66359 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #66359 | C2 |
| Frattini | C32 | C3⋊S4 |
| Fitting | C22×C3×C9 | D3 |
| socle | C22×C3 | #272 |
| LCS | G ▷ #66359 | |
| UCS | C1 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3^4,2^2|b^3=e^2,c^3=e^2,b^a=b^2e,c^a=c^2e,d^a=de^2,e^a=e^2,f^a=g,g^a=f,c^b=cd,d^b=de,f^b=g,g^b=fg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#66359⋊C2, #11757398⋊C2, #11757420⋊C2, … |
| maximal normal subgroup of | C2×#66359⋊C2, #11757398⋊C2, #11757420⋊C2, … |
| maximal quotient of | #13046989, #13046993, #66359⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 271);