Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | A4⋊(A4⋊C8) |
| derived | A42 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4⋊S4 |
| commutator, hypocenter | A42 | C8 |
| Fitting | C24×C4 | C3⋊D3 |
| socle | C25 | C3⋊Dic3 |
| prime radical | C2×A42 | C4 |
| cosocle | C4×A42 | C2 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C4 | |
268 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 34 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:
2^3,3^2,2^4|a^2=b,b^2=c,d^a=d^2,e^a=e^2,f^a=fg,h^a=hi,h^d=i,i^d=hi,f^e=g,g^e=fg56 conjugacy classes — view the full character table.
| maximal subgroup of | C8×A4⋊S4, #72228069, #72228080, … |
| maximal normal subgroup of | C8×A4⋊S4, #72228069, #72228080, … |
| maximal quotient of | C2×A4⋊(A4⋊C8), #72253098 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155479);