Finite Group DatabasebetaFinite Group Databasebeta| order | 528 |
| isoclinism | isoclinic class of D33 |
| automorphism | Aut(G) = C2×D3×D4×AGL(1,𝔽11) (order 10560) · Inn(G) = D33 · Out(G) = C22×C5×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊(C11⋊C8) |
| unambiguous semidirect product | (C2×C3)⋊(C11⋊C8) |
| unambiguous semidirect product | (C2×C11)⋊(C3⋊C8) |
| derived, Hall | C3×C11 ⋊ C2×C8 |
| Sylow | C11 ⋊ C2×C3⋊C8 |
| Sylow | C3 ⋊ C2×C11⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | D33 |
| commutator, hypocenter | C3×C11 | C2×C8 |
| Frattini | C4 | C2×D33 |
| Fitting | C2×C3×C4×C11 | C2 |
| socle, prime radical | C22×C3×C11 | C4 |
| cosocle | C3×C4×C11 | C22 |
| LCS | G ▷ C3×C11 | |
| UCS | C1 ◁ C2×C4 | |
44 conjugacy classes of subgroups.
| transitive degree | μt(G) = 528 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×Dic11, Dic3×C11⋊C8, #11760489, … |
| maximal normal subgroup of | C3⋊C8×Dic11, Dic3×C11⋊C8, #11760489, … |
| maximal quotient of | C4×C3⋊(C11⋊C8), C4⋊(C3⋊(C11⋊C8)), C2×C3⋊(C11⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(528, 68);