Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C3 |
| direct product | C4×A4×SL(2,𝔽3) |
| unambiguous semidirect product | (C4×Q8×A4)⋊C3 |
| unambiguous semidirect product | (C22×C4×SL(2,𝔽3))⋊C3 |
| unambiguous semidirect product | (C4×C22⋊SL(2,𝔽3))⋊C3 |
| Fitting, Sylow | C22×C4×Q8 ⋊ C32 |
| derived | C22×Q8 ⋊ C32×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | A42 |
| commutator, hypocenter | C22×Q8 | C32×C4 |
| Frattini | C22 | C2×A42 |
| Fitting | C22×C4×Q8 | C32 |
| socle | C24 | C2×C3×A4 |
| prime radical | C2×A4×SL(2,𝔽3) | C2 |
| cosocle | C23×Q8 | C2×C32 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C2×C4 | |
327 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 48 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, 156565);