Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×A4×#226 |
| unambiguous semidirect product | (C23×#226)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×S4 |
| commutator, cosocle, hypocenter | C22×SL(2,𝔽3) | C22×C3 |
| Frattini | C2 | C2×A4×S4 |
| Fitting | C23×Q8 | C3×D3 |
| socle | C24 | C3×S4 |
| prime radical | C2×A4×SL(2,𝔽3) | C2 |
| LCS | G ▷ C22×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
415 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 415 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 |
64 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 157488);