Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×GL(2,ℤ/4) |
| unambiguous semidirect product | (C22×GL(2,ℤ/4))⋊C3 |
| Fitting | C26 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×A4×S4 |
| commutator, cosocle | C23×A4 | C22×C3 |
| Fitting | C26 | C3×D3 |
| socle | C25 | C2×C3×D3 |
| hypercenter | C22 | A4×S4 |
| hypocenter | C22×A4 | C3×D4 |
| LCS | G ▷ C23×A4 ▷ C22×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
772 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 772 conjugacy classes of subgroups — too many to draw. Only its 36 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, A4×A4⋊S4, S4×GL(2,ℤ/4), … |
| maximal normal subgroup of | #72215293, S4×GL(2,ℤ/4), A4×#5912, … |
| maximal quotient of | A4×#5414, A4×#5415, #72225978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157506);