Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| direct product | SL(2,𝔽3)×SL(2,ℤ/4) |
| unambiguous semidirect product | A4⋊(C4×SL(2,𝔽3)) |
| unambiguous semidirect product | (Q8×SL(2,ℤ/4))⋊C3 |
| derived | Q8×A4 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | A4×S4 |
| commutator, hypocenter | Q8×A4 | C3×C4 |
| Fitting | C23×Q8 | C3×D3 |
| socle | C24 | D3×A4 |
| prime radical | C2×A4×SL(2,𝔽3) | C2 |
| cosocle | C2×Q8×A4 | C2×C3 |
| LCS | G ▷ Q8×A4 | |
| UCS | C1 ◁ C22 | |
271 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 271 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | GL(2,𝔽3)×SL(2,ℤ/4), SL(2,𝔽3)×GL(2,ℤ/4), #72244327, … |
| maximal normal subgroup of | GL(2,𝔽3)×SL(2,ℤ/4), SL(2,𝔽3)×GL(2,ℤ/4), #72244327 |
| maximal quotient of | SL(2,𝔽3)×GU(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 155461);