Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10620193 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C34 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34 | C8 |
| prime radical | C3⋊(C3⋊(C3⋊D3)) | C4 |
| cosocle | #66472 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^a=d^2e^2,e^a=de^2,f^a=f^2g^2,g^a=f^2,d^b=e^2,e^b=d,f^b=f^2g,g^b=fg,d^c=d^2,e^c=e^2,f^c=f^2,g^c=g^218 conjugacy classes — view the full character table.
| maximal subgroup of | #13049788, #13049790, #13049791, … |
| maximal normal subgroup of | #13049788, #13049790, #13049791, … |
| maximal quotient of | #13049380, C2×#10620193, #14519083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 712);