Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of S42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| direct product | S4×SL(2,ℤ/4) |
| unambiguous semidirect product | (A4×S4)⋊C4 |
| derived | A42 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | S42 |
| commutator, hypocenter | A42 | C2×C4 |
| Fitting, socle | C25 | D32 |
| prime radical | C2×A4×S4 | C2 |
| cosocle | C2×A42 | C22 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C2 | |
573 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 573 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | S4×GL(2,ℤ/4), #72225971, #72228048, … |
| maximal normal subgroup of | S4×GL(2,ℤ/4), #72225971, #72228048, … |
| maximal quotient of | S4×A4⋊AGL(1,𝔽5), GL(2,𝔽3)×SL(2,ℤ/4), S4×GU(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157468);