Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | #737 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C3×C8 |
| Frattini, socle | C2×C3 | C3×#157 |
| Fitting | C2×#737 | C4 |
| prime radical | C2×H(3,𝔽3) | C3×C4 |
| cosocle | #1084 | C2×C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C9 | |
46 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3,2^2,3^3|a^2=c,b^3=g^2,c^2=d,e^a=ef,f^a=ef^2g^2,e^c=e^2,f^c=f^2g,f^e=fg84 conjugacy classes — view the full character table.
| maximal subgroup of | #13048221, #13048270, #13048271, … |
| maximal normal subgroup of | #13048221, #13048270, #13048271, … |
| maximal quotient of | #13047521, C2×#10619833, C9×#6188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 352);