Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #737 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | PSU(3,𝔽2) |
| commutator, prime radical, cosocle | #272 | C22×C3 |
| Frattini, socle | C3 | C3×PSU(3,𝔽2) |
| Fitting | #737 | Q8 |
| hypocenter | H(3,𝔽3) | C3×Q8 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C9 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
2^2,3,2,3^3|a^2=d,b^2=d,c^3=g^2,b^a=bd,e^a=fg^2,f^a=e^2,e^b=e^2f^2,f^b=e^2fg^2,e^d=e^2g^2,f^d=f^2g,f^e=fg48 conjugacy classes — view the full character table.
| maximal subgroup of | #10620020⋊C2, C2×#10620032, #14517899, … |
| maximal normal subgroup of | #10620020⋊C2, C2×#10620032, #14517899, … |
| maximal quotient of | #13048277, C2×#10620032, C9×SU(3,𝔽2), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 551);