Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | #127⋊C27 |
| derived | C42 ⋊ C2×C27 |
| Sylow | #127 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #127⋊C3 |
| commutator, hypocenter | C42 | C2×C27 |
| Frattini | C22×C9 | C2×A4 |
| Fitting | C9×#127 | C3 |
| socle | C22×C3 | C2×C22⋊C9 |
| prime radical | C3×#127 | C9 |
| cosocle | C42×C9 | C2×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C9 | |
72 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3^3,2^4|b^3=c,c^3=d,e^2=g,f^2=h,e^a=eg,f^a=fh,e^b=fg,f^b=ef,g^b=h,h^b=gh90 conjugacy classes — view the full character table.
| maximal subgroup of | #13134131, #11720106⋊C2, C2×#127⋊C27, … |
| maximal normal subgroup of | #13134131, #11720106⋊C2, C2×#127⋊C27, … |
| maximal quotient of | #384⋊C27, #392⋊C27, #399⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 81);