Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #126⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #126⋊C27 |
| derived | C42 ⋊ C2×C27 |
| Sylow | #126 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #126⋊C3 |
| commutator, hypocenter | C42 | C2×C27 |
| Frattini | C22×C9 | C2×A4 |
| Fitting | C9×#126 | C3 |
| socle | C22×C3 | C2×C22⋊C9 |
| prime radical | C23×C3 | C22⋊C9 |
| cosocle | C42×C9 | C2×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C9 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^4|b^3=c,c^3=d,e^2=g,f^2=h,e^a=egh,f^a=fg,e^b=fg,f^b=ef,g^b=h,h^b=gh90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#126⋊C27, #560⋊C27 |
| maximal normal subgroup of | C2×#126⋊C27, #560⋊C27 |
| maximal quotient of | #383⋊C27, #396⋊C27, #401⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 80);