Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊GU(2,𝔽3))⋊C3 |
| derived | C3×SL(2,𝔽3) ⋊ C3×C4 |
104 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2,3^2,2^3|a^2=c,c^2=h,f^2=h,g^2=h,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,d^b=de,f^d=g,g^d=fgh,g^f=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #13146373, #13146374, #13146376, … |
| maximal normal subgroup of | #13146373, #13146374, #13146376, … |
| maximal quotient of | (C3⋊(SL(2,𝔽3)⋊C8))⋊C3, C2×(C3⋊GU(2,𝔽3))⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 695);