Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D3×D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | C3×C9 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D3×D9 |
| commutator, hypocenter | C3×C9 | C4×C8 |
| Frattini | C2×C3×C4 | D32 |
| Fitting, cosocle | C2×C3×C4×C9 | C22 |
| socle, prime radical | C22×C32 | C4×D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C2×C4 | |
162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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,f^3=h,g^a=g^2,f^b=f^2h^2,h^b=h^2144 conjugacy classes — view the full character table.
| maximal subgroup of | #13137063, #13137064, #13137065, … |
| maximal normal subgroup of | #13137063, #13137064, #13137065, … |
| maximal quotient of | C3⋊C8×C9⋊C8, #13134922, #13135012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 238);