Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C3×C9 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×S4 |
| Fitting | C22×C3×C9 | D3 |
| socle | C22×C3 | (C3⋊D3)⋊C3 |
| prime radical | #6258 | C3 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3^4,2^2|b^3=e,c^a=c^2,d^a=d^2e^2,f^a=g,g^a=f,c^b=cd,d^c=de^2,f^c=g,g^c=fg49 conjugacy classes — view the full character table.
| maximal subgroup of | #10619741⋊C2, C2×#10619741, C3×#10619741, … |
| maximal normal subgroup of | #10619741⋊C2, C2×#10619741, C3×#10619741, … |
| maximal quotient of | #13046953, #13046957, #13046971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 260);