Finite Group DatabasebetaFinite Group Databasebeta| order | 220 |
| isoclinism | isoclinic class of C11⋊C5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = C11⋊C5 · Out(G) = C22 |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C4×C11⋊C5 |
| unambiguous semidirect product | (C4×C11)⋊C5 |
| Fitting, Hall | C4×C11 ⋊ C5 |
| derived, Sylow | C11 ⋊ C4×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C11⋊C5 |
| commutator, hypocenter | C11 | C4×C5 |
| Frattini | C2 | C2×C11⋊C5 |
| Fitting | C4×C11 | C5 |
| socle, cosocle | C2×C11 | C2×C5 |
| prime radical | C2×C11⋊C5 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C4 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 44 |
A minimal faithful permutation representation — 2 generators on 15 points:
(2,3,5,8,7)(4,6,10,11,9)(12,13,14,15);(1,2,4,7,3,5,9,11,6,8,10)28 conjugacy classes — view the full character table.
| maximal subgroup of | (C11⋊C5)⋊C8, C8×C11⋊C5, (C11⋊C5)⋊Q8, … |
| maximal normal subgroup of | (C11⋊C5)⋊C8, C8×C11⋊C5, (C11⋊C5)⋊Q8, … |
| maximal quotient of | C8×C11⋊C5, C2×C4×C11⋊C5, Dic3×C11⋊C5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(220, 2);