Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #6293⋊C3 |
| Sylow | C3×H(3,𝔽3) ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C33 | C22×C3 |
| Frattini | C2×C3 | C3×D32 |
| Fitting, prime radical | C2×C3×H(3,𝔽3) | C22 |
| socle | C2×C32 | C2×C3×D3 |
| hypocenter | C33 | C3×Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 | |
131 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 131 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3,2,3^3|a^2=d,b^2=d,b^a=bd,f^a=f^2,e^b=e^2,g^b=g^2,e^c=eg51 conjugacy classes — view the full character table.
| maximal subgroup of | #13048322, #13048323, #13048333, … |
| maximal normal subgroup of | #13048322, #13048323, #13048333, … |
| maximal quotient of | #88807⋊C3, #88806⋊C3, Dic3⋊((C3⋊Dic3)⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 367);