Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C3⋊D3)⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C22×H(3,𝔽3) | C2×C3 |
| Frattini | C32 | D3×A4 |
| socle | C22×C3 | (C3⋊D3)⋊C3 |
| LCS | G ▷ C22×H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
90 conjugacy classes of subgroups.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3,2^2,3^3|b^3=g,e^a=e^2,f^a=f^2g,c^b=cd,d^b=c,e^b=ef,f^e=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #10619743⋊C2, C2×#10619771, #14518148, … |
| maximal normal subgroup of | #10619743⋊C2, C2×#10619771, #14518148, … |
| maximal quotient of | #13047046, #13047048, #13047058, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 290);