Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | 2−1+4⋊C27 |
| Sylow | 2−1+4 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C9 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C27 |
| Fitting | C9×2−1+4 | C3 |
| socle | C2×C3 | C22×C22⋊C9 |
| prime radical | C3×2−1+4 | C9 |
| cosocle | Q8×C9 | C22×C3 |
| hypercenter | D4×C9 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C9 ◁ D4×C9 | |
112 conjugacy classes of subgroups.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^2,3^3,2^3|c^3=d,d^3=e,f^2=h,g^2=h,b^a=bh,f^c=g,g^c=fgh,g^f=gh171 conjugacy classes — view the full character table.
| maximal subgroup of | #13136168, #13136169, #577⋊C27, … |
| maximal normal subgroup of | #13136168, #13136169, #577⋊C27, … |
| maximal quotient of | #519⋊C27, #520⋊C27, #548⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 628);