Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D9×S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C9⋊S4)⋊C4 |
| derived | C9×A4 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×S4 |
| commutator, hypocenter | C9×A4 | C2×C4 |
| Frattini | C2×C3 | D3×S4 |
| Fitting | C23×C9 | C2×D3 |
| socle | C23×C3 | D32 |
| prime radical | C2×C9⋊S4 | C2 |
| cosocle | C2×C9×A4 | C22 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C2 | |
188 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^3,2^2|a^2=c,b^2=c,e^3=f,e^a=e^2f^2,f^a=f^2,d^b=d^2,g^b=h,h^b=g,g^d=h,h^d=gh60 conjugacy classes — view the full character table.
| maximal subgroup of | #13164104, (C9⋊S4)⋊Q8, #13164108, … |
| maximal normal subgroup of | #13164104, (C9⋊S4)⋊Q8, #13164108, … |
| maximal quotient of | (C9⋊S4)⋊C8, #13146192, #13146207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2207);