Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C3⋊#63421 |
| derived | C3×Q8⋊C9 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊((C22⋊C9)⋊C2) |
| commutator, hypocenter | C3×Q8⋊C9 | C4 |
| Frattini | C22×C3 | C3⋊S4 |
| Fitting | C2×C32×Q8 | D3 |
| socle, prime radical | C22×C32 | S4 |
| cosocle | C2×C3×Q8⋊C9 | C2 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C22 | |
146 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,c^3=e^2,f^2=h,g^2=h,c^a=c^2e,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh72 conjugacy classes — view the full character table.
| maximal subgroup of | #13146299, Dic3×#3688, #13146302, … |
| maximal normal subgroup of | #13146299, Dic3×#3688, #13146302, … |
| maximal quotient of | C3⋊((Q8⋊C9)⋊C8), C2×C3⋊#63421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 704);