Finite Group DatabasebetaFinite Group Databasebeta| order | 220 |
| isoclinism | isoclinic class of D55 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×AGL(1,𝔽11) (order 4400) · Inn(G) = D55 · Out(G) = C2×C4×C5 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D110 |
| direct product | C2×D55 |
| unambiguous semidirect product | (C2×C5)⋊D11 |
| unambiguous semidirect product | (C2×C11)⋊D5 |
| Fitting, socle | C2×C5×C11 ⋊ C2 |
| derived, Hall | C5×C11 ⋊ C22 |
| Sylow | C11 ⋊ C2×D5 |
| Sylow | C5 ⋊ C2×D11 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D55 |
| commutator, cosocle, hypocenter | C5×C11 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C5×C11 | C2 |
| LCS | G ▷ C5×C11 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 110 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(3,4,5,6,7)(8,9,10,11,12,13,14,15,16,17,18);(4,7)(5,6)(9,18)(10,17)(11,16)(12,15)(13,14)58 conjugacy classes — view the full character table.
| maximal subgroup of | #89166, #89168, #89169, … |
| maximal normal subgroup of | #89166, #89168, #89169, … |
| maximal quotient of | Dic110, C4×D55, D220, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(220, 14);