Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C36 |
No aliases for this group.
| Fitting | C22×C34 ⋊ C2 |
| Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊(C3⋊D3)) |
| commutator, socle, cosocle | C2×C34 | C22 |
| Fitting | C22×C34 | C2 |
| hypercenter | C22 | C3⋊(C3⋊(C3⋊D3)) |
| hypocenter | C34 | D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1696 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^3,3^4|b^a=bc,d^a=d^2,e^a=e^2,f^a=f^2,g^a=g^2165 conjugacy classes — view the full character table.
| maximal subgroup of | #13049674, #13049677, #67×C3⋊(C3⋊D3), … |
| maximal normal subgroup of | #13049674, #13049677, #67×C3⋊(C3⋊D3), … |
| maximal quotient of | #13049155, #13049157, #13049158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 698);