Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C27 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #131⋊C27 |
| Sylow | #131 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C9 | A4 |
| commutator, hypocenter | Q8 | C4×C27 |
| Frattini | C4×C9 | C2×A4 |
| Fitting | C9×#131 | C3 |
| socle | C2×C3 | C4×C22⋊C9 |
| prime radical | C3×#41 | C2×C9 |
| cosocle | C9×#41 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C8×C9 | |
52 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2,3,2,3^2,2^3|a^2=c,b^3=d,c^2=h,d^3=e,f^2=h,g^2=h,f^b=g,g^b=fgh,g^f=gh252 conjugacy classes — view the full character table.
| maximal subgroup of | #13134133, #504⋊C27, #13136141, … |
| maximal normal subgroup of | #13134133, #504⋊C27, #13136141, … |
| maximal quotient of | C8×Q8⋊C27, C2×#131⋊C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 83);