Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | C9⋊#859 |
| derived | C9×SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C9⋊S4 |
| commutator, hypocenter | C9×SL(2,𝔽3) | C4 |
| Frattini, socle | C22×C3 | C3⋊S4 |
| Fitting | C2×Q8×C9 | D3 |
| prime radical | C2×C3×SL(2,𝔽3) | D3 |
| cosocle | C2×C9×SL(2,𝔽3) | C2 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
128 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=b,d^3=e^2,f^2=h,g^2=h,c^a=c^2,d^a=d^2e,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 | #13146205, Dic9×#226, #13146208, … |
| maximal normal subgroup of | #13146205, Dic9×#226, #13146208, … |
| maximal quotient of | C9⋊(SL(2,𝔽3)⋊C8), C2×C9⋊#859 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 698);