Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #63421⋊C3 |
| derived | Q8⋊C9 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C22⋊C9)⋊C2)⋊C3 |
| commutator, hypocenter | Q8⋊C9 | C3×C4 |
| Frattini, socle | C22×C3 | C3×S4 |
| Fitting | C2×C32×Q8 | D3 |
| prime radical | C22×C32 | S4 |
| cosocle | C2×Q8⋊C9 | C2×C3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C22 | |
101 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2,3,2,3^2,2^3|a^2=c,d^3=e^2,f^2=h,g^2=h,d^a=d^2e,e^a=e^2,f^a=gh,g^a=fh,d^b=de,f^d=g,g^d=fgh,g^f=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #10589488⋊C3, C4×#3688⋊C3, #10589490⋊C3, … |
| maximal normal subgroup of | #10589488⋊C3, C4×#3688⋊C3, #10589490⋊C3, … |
| maximal quotient of | #6204⋊C8, C2×#63421⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 690);