Finite Group DatabasebetaFinite Group Databasebeta| order | 220 |
| isoclinism | isoclinic class of C11⋊AGL(1,𝔽5) |
| automorphism | Aut(G) = AGL(1,𝔽5)×AGL(1,𝔽11) (order 2200) · Inn(G) = G · Out(G) = C2×C5 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C11⋊AGL(1,𝔽5) |
| Fitting, socle, derived, Hall | C5×C11 ⋊ C4 |
| Sylow | C11 ⋊ AGL(1,𝔽5) |
| Sylow | C5 ⋊ Dic11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C5×C11 | C4 |
| prime radical, cosocle | D5×C11 | C2 |
| LCS | G ▷ C5×C11 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 55 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(13,14,15,16);(1,2,4,6,8,10,11,9,7,5,3)(12,13,16,14,15)Polycyclic presentation:
2^2,5,11|a^2=b,c^a=c^3,d^a=d^10,c^b=c^425 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(220, 10);