Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | ((C3⋊D3)⋊A4)⋊C4 |
| derived | C22⋊H(3,𝔽3) ⋊ C2×C4 |
267 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 267 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|b^2=c,e^a=e^2,f^a=f^2,d^b=d^2,f^b=f^2,g^b=h,h^b=g,e^d=ef,g^d=h,h^d=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊A4)⋊Q8, #13164169, #13164170, … |
| maximal normal subgroup of | ((C3⋊D3)⋊A4)⋊Q8, #13164169, #13164170, … |
| maximal quotient of | ((C3⋊D3)⋊A4)⋊C8, #13146284, #13146327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2235);