Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D32⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D32⋊S4 |
| Fitting, socle | C22×C32 ⋊ #67 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | A4×C3⋊D3 | C22 |
| Fitting, socle | C22×C32 | #67 |
| hypocenter | C32×A4 | D4 |
| LCS | G ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 | |
212 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3^3,2^2|b^a=bc,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,e^b=e^2,e^c=e^2,f^c=f^2,g^d=gh,h^d=g27 conjugacy classes — view the full character table.
| maximal subgroup of | S4×GO+(4,𝔽2), C2×D32⋊S4, #72241538, … |
| maximal normal subgroup of | S4×GO+(4,𝔽2), C2×D32⋊S4 |
| maximal quotient of | #13180205, #13180206, #13180207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4670);