Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#6188 |
| derived, Sylow | C3×H(3,𝔽3) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊#157 |
| commutator, hypocenter | C3×H(3,𝔽3) | C8 |
| Fitting, prime radical | C2×C3×H(3,𝔽3) | C4 |
| socle | C2×C32 | #157 |
| cosocle | C3×#1084 | C2 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
75 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3^4|a^2=b,b^2=c,d^a=eg^2,e^a=d^2,f^a=f^2,d^b=d^2g^2,e^b=e^2g,e^d=eg48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#6188, #13048233, #13048234, … |
| maximal normal subgroup of | D3×#6188, #13048233, #13048234, … |
| maximal quotient of | C3⋊#88410, C2×C3⋊#6188, C9⋊#6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 354);