Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | A4⋊GU(2,𝔽3) |
| derived | A4×SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4⋊S4 |
| commutator, hypocenter | A4×SL(2,𝔽3) | C4 |
| Fitting | C22×#41 | C3⋊D3 |
| socle | C23 | C3⋊SL(2,ℤ/4) |
| prime radical, cosocle | A4×#41⋊C3 | C2 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
237 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 237 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.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3^2,2^5|a^2=b,b^2=i,g^2=i,h^2=i,c^a=c^2,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,g^c=h,h^c=ghi,e^d=f,f^d=ef,h^g=hi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72275838, S4×GU(2,𝔽3) |
| maximal normal subgroup of | S4×GU(2,𝔽3) |
| maximal quotient of | #72276811 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155481);