Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #63690⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C22×C3 |
| Frattini, socle | C2×C3 | C2×C3×S4 |
| Fitting, prime radical | C32×#41 | D3 |
| hypercenter | C4 | ((C22⋊C9)⋊C2)⋊C3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
132 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=h,b^2=h,d^3=e,f^2=h,g^2=h,b^a=bh,d^a=d^2e^2,e^a=e^2,f^a=gh,g^a=fh,d^c=de^2,f^d=g,g^d=fgh,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #10589505⋊C3, #10589508⋊C3, #10589530⋊C3, … |
| maximal normal subgroup of | #10589505⋊C3, #10589508⋊C3, #10589530⋊C3, … |
| maximal quotient of | #10589488⋊C3, #10589490⋊C3, C4⋊(#3688⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2318);