Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#41⋊C3 |
| unambiguous semidirect product | #41⋊(C3×C11) |
| unambiguous semidirect product | (C11×#41)⋊C3 |
| Fitting, Hall | C11×#41 ⋊ C3 |
| Sylow | #41 ⋊ C3×C11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C11 | A4 |
| commutator, cosocle, hypocenter | Q8 | C2×C3×C11 |
| Frattini | C2 | C2×C11×A4 |
| Fitting | C11×#41 | C3 |
| socle | C2×C11 | C2×A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4×C11 | |
30 conjugacy classes of subgroups.
| transitive degree | μt(G) = 176 |
154 conjugacy classes — view the full character table.
| maximal subgroup of | C11×GU(2,𝔽3), C11⋊GU(2,𝔽3), C11⋊(#131⋊C3), … |
| maximal normal subgroup of | C11×GU(2,𝔽3), C11⋊GU(2,𝔽3), C11⋊(#131⋊C3), … |
| maximal quotient of | C4×C11×SL(2,𝔽3), C2×C11×#41⋊C3, C11×#41⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 125);