Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C2×D4×C27 ⋊ C2 |
| Sylow | C27 ⋊ #127 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D27 |
| commutator, cosocle | C22×C27 | C23 |
| Frattini | C22×C9 | C22×D3 |
| Fitting | C2×D4×C27 | C2 |
| socle | C22×C3 | C22×D9 |
| hypercenter | C2×D4 | D27 |
| hypocenter | C27 | #127 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|b^2=e,f^3=g,g^3=h,b^a=be,c^a=cd,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=ce144 conjugacy classes — view the full character table.
| maximal subgroup of | #13133970, #13134252, #13134262, … |
| maximal normal subgroup of | #13133970, #13134252, #13134262, … |
| maximal quotient of | #13134448, #13134456, #13134465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 156);