Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C36 |
| unambiguous semidirect product | C4⋊(C3⋊(C3⋊(C3⋊D3))) |
| Fitting | C34×C4 ⋊ C2 |
| Sylow | C34 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊(C3⋊D3)) |
| commutator, socle, cosocle | C2×C34 | C22 |
| Fitting | C34×C4 | C2 |
| hypercenter | C4 | C3⋊(C3⋊(C3⋊D3)) |
| hypocenter | C34 | D4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 ◁ C4 | |
1696 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2^3,3^4|b^2=c,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 | #13049048, #13049049, #13049053, … |
| maximal normal subgroup of | #13049048, #13049049, #13049053, … |
| maximal quotient of | #13049149, #13049150, #13049151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 696);