Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D3×D9 |
| commutator | C2×C3×C9 | C42 |
| Frattini | C2×C3×C4 | D32 |
| Fitting, cosocle | C2×C3×C4×C9 | C22 |
| socle, prime radical | C22×C32 | C2×Dic3 |
| hypocenter | C3×C9 | #97 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2×C4 | |
142 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 67 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=d,b^2=c,c^2=e,f^3=h,b^a=be,g^a=g^2,f^b=f^2h^2,h^b=h^2132 conjugacy classes — view the full character table.
| maximal subgroup of | #13137021, #13137022, #13137023, … |
| maximal normal subgroup of | #13137021, #13137022, #13137023, … |
| maximal quotient of | #13134921, #13134922, #13134988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 225);