Properties

 Label 20T1 Degree $20$ Order $20$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_{20}$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 1);

Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{20}$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $20$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,12,2,11)(3,14,4,13)(5,15,6,16)(7,17,8,18)(9,20,10,19), (1,4,6,8,9,12,13,16,18,20,2,3,5,7,10,11,14,15,17,19) magma: Generators(G);

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 10: $C_{10}$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

 Label Cycle Type Size Order Representative $1^{20}$ $1$ $1$ $()$ $2^{10}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ $20$ $1$ $20$ $( 1, 3, 6, 7, 9,11,13,15,18,19, 2, 4, 5, 8,10,12,14,16,17,20)$ $20$ $1$ $20$ $( 1, 4, 6, 8, 9,12,13,16,18,20, 2, 3, 5, 7,10,11,14,15,17,19)$ $5^{4}$ $1$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$ $10^{2}$ $1$ $10$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$ $20$ $1$ $20$ $( 1, 7,13,19, 5,12,17, 3, 9,15, 2, 8,14,20, 6,11,18, 4,10,16)$ $20$ $1$ $20$ $( 1, 8,13,20, 5,11,17, 4, 9,16, 2, 7,14,19, 6,12,18, 3,10,15)$ $5^{4}$ $1$ $5$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$ $10^{2}$ $1$ $10$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$ $4^{5}$ $1$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$ $4^{5}$ $1$ $4$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$ $10^{2}$ $1$ $10$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$ $5^{4}$ $1$ $5$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$ $20$ $1$ $20$ $( 1,15,10, 3,18,12, 6,19,14, 7, 2,16, 9, 4,17,11, 5,20,13, 8)$ $20$ $1$ $20$ $( 1,16,10, 4,18,11, 6,20,14, 8, 2,15, 9, 3,17,12, 5,19,13, 7)$ $10^{2}$ $1$ $10$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$ $5^{4}$ $1$ $5$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$ $20$ $1$ $20$ $( 1,19,17,15,14,11,10, 7, 5, 3, 2,20,18,16,13,12, 9, 8, 6, 4)$ $20$ $1$ $20$ $( 1,20,17,16,14,12,10, 8, 5, 4, 2,19,18,15,13,11, 9, 7, 6, 3)$

magma: ConjugacyClasses(G);

Group invariants

 Order: $20=2^{2} \cdot 5$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 20.2 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9 Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 P 1A 1A 2A 2A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 10A3 10A1 10A3 10A-3 10A-3 10A-1 10A-1 10A1 5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A-1 4A1 4A-1 4A1 Type 20.2.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 20.2.1b R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ 20.2.1c1 C $1$ $−1$ $−i$ $i$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−i$ $i$ $i$ $−i$ $i$ $−i$ $−i$ $i$ 20.2.1c2 C $1$ $−1$ $i$ $−i$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $i$ $−i$ $−i$ $i$ $−i$ $i$ $i$ $−i$ 20.2.1d1 C $1$ $1$ $1$ $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ 20.2.1d2 C $1$ $1$ $1$ $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ 20.2.1d3 C $1$ $1$ $1$ $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ 20.2.1d4 C $1$ $1$ $1$ $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ 20.2.1e1 C $1$ $1$ $−1$ $−1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $−ζ5−1$ $−ζ5$ $−ζ5−2$ $−ζ52$ 20.2.1e2 C $1$ $1$ $−1$ $−1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $−ζ5$ $−ζ5−1$ $−ζ52$ $−ζ5−2$ 20.2.1e3 C $1$ $1$ $−1$ $−1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $−ζ52$ $−ζ5−2$ $−ζ5−1$ $−ζ5$ 20.2.1e4 C $1$ $1$ $−1$ $−1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $−ζ5−2$ $−ζ52$ $−ζ5$ $−ζ5−1$ 20.2.1f1 C $1$ $−1$ $−ζ205$ $ζ205$ $−ζ202$ $ζ208$ $ζ204$ $−ζ206$ $ζ206$ $−ζ204$ $−ζ208$ $ζ202$ $ζ203$ $−ζ207$ $ζ209$ $−ζ20$ $ζ20$ $−ζ209$ $ζ207$ $−ζ203$ 20.2.1f2 C $1$ $−1$ $ζ205$ $−ζ205$ $ζ208$ $−ζ202$ $−ζ206$ $ζ204$ $−ζ204$ $ζ206$ $ζ202$ $−ζ208$ $−ζ207$ $ζ203$ $−ζ20$ $ζ209$ $−ζ209$ $ζ20$ $−ζ203$ $ζ207$ 20.2.1f3 C $1$ $−1$ $−ζ205$ $ζ205$ $ζ208$ $−ζ202$ $−ζ206$ $ζ204$ $−ζ204$ $ζ206$ $ζ202$ $−ζ208$ $ζ207$ $−ζ203$ $ζ20$ $−ζ209$ $ζ209$ $−ζ20$ $ζ203$ $−ζ207$ 20.2.1f4 C $1$ $−1$ $ζ205$ $−ζ205$ $−ζ202$ $ζ208$ $ζ204$ $−ζ206$ $ζ206$ $−ζ204$ $−ζ208$ $ζ202$ $−ζ203$ $ζ207$ $−ζ209$ $ζ20$ $−ζ20$ $ζ209$ $−ζ207$ $ζ203$ 20.2.1f5 C $1$ $−1$ $−ζ205$ $ζ205$ $−ζ206$ $ζ204$ $−ζ202$ $ζ208$ $−ζ208$ $ζ202$ $−ζ204$ $ζ206$ $−ζ209$ $ζ20$ $−ζ207$ $ζ203$ $−ζ203$ $ζ207$ $−ζ20$ $ζ209$ 20.2.1f6 C $1$ $−1$ $ζ205$ $−ζ205$ $ζ204$ $−ζ206$ $ζ208$ $−ζ202$ $ζ202$ $−ζ208$ $ζ206$ $−ζ204$ $ζ20$ $−ζ209$ $ζ203$ $−ζ207$ $ζ207$ $−ζ203$ $ζ209$ $−ζ20$ 20.2.1f7 C $1$ $−1$ $−ζ205$ $ζ205$ $ζ204$ $−ζ206$ $ζ208$ $−ζ202$ $ζ202$ $−ζ208$ $ζ206$ $−ζ204$ $−ζ20$ $ζ209$ $−ζ203$ $ζ207$ $−ζ207$ $ζ203$ $−ζ209$ $ζ20$ 20.2.1f8 C $1$ $−1$ $ζ205$ $−ζ205$ $−ζ206$ $ζ204$ $−ζ202$ $ζ208$ $−ζ208$ $ζ202$ $−ζ204$ $ζ206$ $ζ209$ $−ζ20$ $ζ207$ $−ζ203$ $ζ203$ $−ζ207$ $ζ20$ $−ζ209$

magma: CharacterTable(G);