Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | A4×#131⋊C3 |
| unambiguous semidirect product | (A4×#131)⋊C3 |
| unambiguous semidirect product | (C22×#131⋊C3)⋊C3 |
| unambiguous semidirect product | #85665⋊C3 |
| Fitting, Sylow | C22×#131 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | A42 |
| commutator, hypocenter | C22×Q8 | C32×C4 |
| Frattini | C4 | C2×A42 |
| Fitting | C22×#131 | C32 |
| socle | C23 | C3×C4×A4 |
| prime radical | A4×#41⋊C3 | C2 |
| cosocle | C22×#41 | C2×C32 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C8 | |
231 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 231 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) = 96 |
112 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 156575);