Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of (C3⋊S4)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | (C3⋊(A4⋊C8))⋊C3 |
| derived | C3×A4 ⋊ C3×C8 |
115 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2^2,3^2,2^2|a^2=c,c^2=d,e^a=e^2,f^a=f^2,g^a=h,h^a=g,e^b=ef,g^e=h,h^e=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊A4)⋊C8, #6258⋊C8, #13146284, … |
| maximal normal subgroup of | ((C3⋊D3)⋊A4)⋊C8, #6258⋊C8, #13146284, … |
| maximal quotient of | (C3⋊(SL(2,𝔽3)⋊C8))⋊C3, (C3⋊(A4⋊C16))⋊C3, (C3⋊#4628)⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 689);