Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4⋊#857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | A4⋊#857 |
| Fitting | C22×C42 ⋊ C3⋊D3 |
| derived | A4×C42⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×C42⋊C3 | C2 |
| Frattini | C22 | A4⋊S4 |
| Fitting | C22×C42 | C3⋊D3 |
| socle | C24 | C3⋊S4 |
| LCS | G ▷ A4×C42⋊C3 | |
| UCS | C1 | |
219 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 219 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) = 36 |
Polycyclic presentation:
2,3^2,2^6|f^2=hi,g^2=i,b^a=b^2,c^a=c^2,d^a=de,f^a=fg,g^a=gi,h^a=hi,f^b=fg,g^b=fi,h^b=i,i^b=hi,d^c=de,e^c=d28 conjugacy classes — view the full character table.
| maximal subgroup of | #72214357, #72219738, S4×#857, … |
| maximal normal subgroup of | #72219738, S4×#857, #72226115 |
| maximal quotient of | #72214357, #72228155, #72228156 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155474);