Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C24×C3 |
| unambiguous semidirect product | A4⋊#1020 |
| Fitting, socle | C26 ⋊ C3⋊D3 |
| derived | A4×C22⋊A4 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×C22⋊A4 | C2 |
| Fitting, socle | C26 | C3⋊D3 |
| LCS | G ▷ A4×C22⋊A4 | |
| UCS | C1 | |
706 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 21 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:
2,3^2,2^6|b^a=b^2,c^a=c^2,d^a=de,f^a=fg,h^a=hi,h^b=hi,i^b=h,d^c=de,e^c=d,f^c=fg,g^c=f28 conjugacy classes — view the full character table.
| maximal subgroup of | #72215732, S4×#1020, #72218837, … |
| maximal normal subgroup of | S4×#1020, #72233459, #72241489, … |
| maximal quotient of | #72218837, #72220401, #72241468, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157861);