# Properties

 Label 20T3 Degree $20$ Order $20$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group no Group: $C_2\times C_{10}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 3);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $3$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2\times C_{10}$ 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: (1,13,5,17,9,2,14,6,18,10)(3,15,8,20,11,4,16,7,19,12), (1,19,18,16,14,11,9,8,5,3)(2,20,17,15,13,12,10,7,6,4) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$10$:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 10: $C_{10}$ x 3

## 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 1A $1^{20}$ $1$ $1$ $()$ 2A $2^{10}$ $1$ $2$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$ 2B $2^{10}$ $1$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$ 2C $2^{10}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ 5A1 $5^{4}$ $1$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$ 5A-1 $5^{4}$ $1$ $5$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$ 5A2 $5^{4}$ $1$ $5$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$ 5A-2 $5^{4}$ $1$ $5$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$ 10A1 $10^{2}$ $1$ $10$ $( 1, 3, 5, 8, 9,11,14,16,18,19)( 2, 4, 6, 7,10,12,13,15,17,20)$ 10A-1 $10^{2}$ $1$ $10$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$ 10A3 $10^{2}$ $1$ $10$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$ 10A-3 $10^{2}$ $1$ $10$ $( 1, 7,14,20, 5,12,18, 4, 9,15)( 2, 8,13,19, 6,11,17, 3,10,16)$ 10B1 $10^{2}$ $1$ $10$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$ 10B-1 $10^{2}$ $1$ $10$ $( 1,15, 9, 4,18,12, 5,20,14, 7)( 2,16,10, 3,17,11, 6,19,13, 8)$ 10B3 $10^{2}$ $1$ $10$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$ 10B-3 $10^{2}$ $1$ $10$ $( 1,19,18,16,14,11, 9, 8, 5, 3)( 2,20,17,15,13,12,10, 7, 6, 4)$ 10C1 $10^{2}$ $1$ $10$ $( 1, 8,14,19, 5,11,18, 3, 9,16)( 2, 7,13,20, 6,12,17, 4,10,15)$ 10C-1 $10^{2}$ $1$ $10$ $( 1,20,18,15,14,12, 9, 7, 5, 4)( 2,19,17,16,13,11,10, 8, 6, 3)$ 10C3 $10^{2}$ $1$ $10$ $( 1, 4, 5, 7, 9,12,14,15,18,20)( 2, 3, 6, 8,10,11,13,16,17,19)$ 10C-3 $10^{2}$ $1$ $10$ $( 1,16, 9, 3,18,11, 5,19,14, 8)( 2,15,10, 4,17,12, 6,20,13, 7)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $20=2^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 20.5 magma: IdentifyGroup(G); Character table:

 1A 2A 2B 2C 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3 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 1A 1A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A-2 5A2 5A2 5A1 5A-1 5A-2 5A-1 5A1 5A2 5 P 1A 2B 2C 2A 1A 1A 1A 1A 2C 2A 2A 2B 2A 2B 2A 2C 2C 2B 2B 2C Type 20.5.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 20.5.1b R $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ $1$ 20.5.1c R $1$ $−1$ $1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ 20.5.1d R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ 20.5.1e1 C $1$ $1$ $1$ $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ 20.5.1e2 C $1$ $1$ $1$ $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ 20.5.1e3 C $1$ $1$ $1$ $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ 20.5.1e4 C $1$ $1$ $1$ $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ 20.5.1f1 C $1$ $−1$ $−1$ $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ 20.5.1f2 C $1$ $−1$ $−1$ $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ 20.5.1f3 C $1$ $−1$ $−1$ $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ 20.5.1f4 C $1$ $−1$ $−1$ $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ 20.5.1g1 C $1$ $−1$ $1$ $−1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ 20.5.1g2 C $1$ $−1$ $1$ $−1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ 20.5.1g3 C $1$ $−1$ $1$ $−1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ 20.5.1g4 C $1$ $−1$ $1$ $−1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ 20.5.1h1 C $1$ $1$ $−1$ $−1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ 20.5.1h2 C $1$ $1$ $−1$ $−1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ 20.5.1h3 C $1$ $1$ $−1$ $−1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ $−ζ5−2$ $−ζ52$ $−ζ5−1$ $−ζ5$ $−ζ5−1$ $−ζ5$ $−ζ52$ $−ζ5−2$ 20.5.1h4 C $1$ $1$ $−1$ $−1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ $−ζ52$ $−ζ5−2$ $−ζ5$ $−ζ5−1$ $−ζ5$ $−ζ5−1$ $−ζ5−2$ $−ζ52$

magma: CharacterTable(G);