Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of #10620197 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C34 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34 | C2×C4 |
| prime radical | C3⋊D32 | C2 |
| cosocle | C32×C3⋊D3 | C22 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
222 conjugacy classes of subgroups.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,d^a=e^2,e^a=d,g^a=g^2,d^b=e,e^b=d^2,f^b=f^2,d^c=d^2,e^c=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | #13049779, D32⋊#157, #13049781, … |
| maximal normal subgroup of | #13049779, D32⋊#157, #13049781, … |
| maximal quotient of | #13049402, #13049403, #13049404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 716);