Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×(C22⋊C9)⋊C2 |
| commutator, cosocle | C2×C3×C22⋊C9 | C22 |
| Frattini | C2×C3 | D3×S4 |
| Fitting, socle | C23×C32 | C2×D3 |
| hypocenter | C3×C22⋊C9 | D4 |
| LCS | G ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9 | |
| UCS | C1 ◁ C2 | |
225 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 225 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^3,2^2|a^2=c,d^3=f,b^a=bc,e^a=e^2,d^b=d^2f^2,f^b=f^2,g^b=h,h^b=g,g^d=h,h^d=gh48 conjugacy classes — view the full character table.
| maximal subgroup of | #13164155, #13164156, #13164161, … |
| maximal normal subgroup of | #13164155, #13164156, #13164161, … |
| maximal quotient of | #13146263, #13146268, #13146269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2231);