Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| isoclinism | isoclinic class of A4⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25×C3 |
| direct product | C22×A4⋊S4 |
| Fitting, socle | C26 ⋊ C3⋊D3 |
| derived | A42 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4⋊S4 |
| commutator, cosocle, hypocenter | A42 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C26 | C3⋊D3 |
| LCS | G ▷ A42 | |
| UCS | C1 ◁ C22 | |
1674 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1674 conjugacy classes of subgroups — too many to draw. Only its 66 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 | #72215293, D4×A4⋊S4, A4×A4⋊S4, … |
| maximal normal subgroup of | #72215293, D4×A4⋊S4, A4×A4⋊S4, … |
| maximal quotient of | D4×A4⋊S4, C2×D3×A4⋊S4, #72224794, … |
| automorphism group of | #41⋊SL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157860);