Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C9 |
No aliases for this group.
| Sylow | C92 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C9⋊D9 |
| commutator, cosocle | C2×C92 | C22 |
| Frattini, socle, prime radical | C2×C32 | C2×C3⋊D3 |
| Fitting | C4×C92 | C2 |
| hypercenter | C4 | C9⋊D9 |
| hypocenter | C92 | Q8 |
| LCS | G ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ C4 | |
138 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^3,3^4|a^2=c,b^2=c,d^3=f,e^3=g,b^a=bc,d^a=d^2f^2,e^a=e^2g^2,f^a=f^2,g^a=g^2165 conjugacy classes — view the full character table.
| maximal subgroup of | #13046367, #13046370, #13046728, … |
| maximal normal subgroup of | #13046367, #13046370, #13046728, … |
| maximal quotient of | #13046766, C4⋊(C9⋊Dic9), C2×#10619651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 170);