Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C4×A4⋊S4 |
| Fitting | C24×C4 ⋊ C3⋊D3 |
| derived | A42 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4⋊S4 |
| commutator, hypocenter | A42 | C2×C4 |
| Frattini | C2 | C2×A4⋊S4 |
| Fitting | C24×C4 | C3⋊D3 |
| socle | C25 | C2×C3⋊D3 |
| prime radical | C2×A4⋊S4 | C2 |
| cosocle | C2×A42 | C22 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C4 | |
598 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 598 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | D4×A4⋊S4, AGL(1,𝔽5)×A4⋊S4, C2×C4×A4⋊S4, … |
| maximal normal subgroup of | D4×A4⋊S4, C2×C4×A4⋊S4, #72225947, … |
| maximal quotient of | AGL(1,𝔽5)×A4⋊S4, C2×C4×A4⋊S4, C4×SO+(4,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157519);