Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C3×C4 |
No aliases for this group.
| derived, Sylow | C33 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×C3⋊D3 |
| commutator, hypocenter | C33 | C4×C8 |
| Fitting, cosocle | C2×C33×C4 | C22 |
| socle, prime radical | C22×C33 | C2×C4 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
384 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=c,b^2=cd,c^2=e,h^a=h^2,f^b=f^2,g^b=g^2144 conjugacy classes — view the full character table.
| maximal subgroup of | #13154818, #13154820, #13154840, … |
| maximal normal subgroup of | #13154818, #13154820, #13154840, … |
| maximal quotient of | C3⋊C8×C3⋊(C3⋊C8), #13145505, #13145544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1985);