Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D32⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D32⋊(C3⋊C8) |
| Sylow | C33 ⋊ C22⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32⋊D3 |
| commutator | C3×C3⋊D3 | C2×C8 |
| Fitting | C33×C4 | D4 |
| socle | C2×C33 | C22⋊C4 |
| prime radical | C2×C3×D32 | C4 |
| cosocle | C3×C4×C3⋊D3 | C22 |
| hypocenter | C33 | C22⋊C8 |
| LCS | G ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C4 | |
200 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|a^2=d,d^2=e,b^a=bc,f^a=f^2,g^a=f^2g,h^a=h^2,f^b=f^2g^2,f^c=f^2,g^c=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D32⋊C8, #13165143, #13165144, … |
| maximal normal subgroup of | D3×D32⋊C8, #13165143, #13165144, … |
| maximal quotient of | D32⋊(C3⋊C16), #13154935, #13154938, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2719);