# Properties

 Label 20T36 Degree $20$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $C_2\times A_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 36);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $36$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2\times A_5$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,20,13)(2,19,14)(3,6,7)(4,5,8)(9,17,12)(10,18,11), (1,20,18,11,8,2,19,17,12,7)(3,16,6,9,14,4,15,5,10,13) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $A_{5}$

## Low degree siblings

10T11, 12T75, 12T76, 20T31, 24T203, 30T29, 30T30, 40T61

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 120.35 magma: IdentifyGroup(G);
 Character table:  2 3 1 3 3 1 3 1 1 1 1 3 1 1 . 1 1 . . . . . 5 1 . . 1 . . 1 1 1 1 1a 3a 2a 2b 6a 2c 5a 5b 10a 10b 2P 1a 3a 1a 1a 3a 1a 5b 5a 5b 5a 3P 1a 1a 2a 2b 2b 2c 5b 5a 10b 10a 5P 1a 3a 2a 2b 6a 2c 1a 1a 2b 2b 7P 1a 3a 2a 2b 6a 2c 5b 5a 10b 10a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 -1 1 1 1 -1 -1 X.3 3 . -1 3 . -1 A *A A *A X.4 3 . -1 3 . -1 *A A *A A X.5 3 . 1 -3 . -1 A *A -A -*A X.6 3 . 1 -3 . -1 *A A -*A -A X.7 4 1 . 4 1 . -1 -1 -1 -1 X.8 4 1 . -4 -1 . -1 -1 1 1 X.9 5 -1 1 5 -1 1 . . . . X.10 5 -1 -1 -5 1 1 . . . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 

magma: CharacterTable(G);