Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C32×C9 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×C3⋊D9 |
| commutator, Fitting, cosocle | C2×C32×C9 | C22 |
| Frattini | C2×C3 | D3×C3⋊D3 |
| socle | C2×C33 | C2×D3 |
| hypocenter | C32×C9 | D4 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
235 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^4|b^2=c,d^3=g,b^a=bc,f^a=f^2,d^b=d^2g^2,e^b=e^2,g^b=g^287 conjugacy classes — view the full character table.
| maximal subgroup of | #13048290, #13048294, D9×#626, … |
| maximal normal subgroup of | #13048290, #13048294, D9×#626, … |
| maximal quotient of | #13047653, #13047657, #13047661, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 399);