Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | A4⋊SL(2,ℤ/4) |
| derived | A42 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊S4 |
| commutator, hypocenter | A42 | C4 |
| Fitting, socle | C25 | C3⋊D3 |
| prime radical, cosocle | C2×A42 | C2 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2 | |
191 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 191 conjugacy classes of subgroups — too many to draw. Only its 23 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:
2^2,3^2,2^4|a^2=b,c^a=c^2,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,g^c=gh,h^c=g,e^d=ef,f^d=e28 conjugacy classes — view the full character table.
| maximal subgroup of | #11923442, S4×SL(2,ℤ/4), #11923459, … |
| maximal normal subgroup of | #11923442, S4×SL(2,ℤ/4), #11923459, … |
| maximal quotient of | A4⋊(A4⋊C8), A4⋊#859, A4⋊GU(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 8283);