# Properties

 Label 20T22 Degree $20$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{10}:C_4$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(20, 22);

## Group action invariants

 Degree $n$: $20$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $22$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_{10}:C_4$ 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: (1,14,5,12)(2,13,6,11)(3,17,4,18)(7,15,9,19)(8,16,10,20), (1,5,10,3,7,2,6,9,4,8)(11,15,20,14,18)(12,16,19,13,17) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_5$

## Low degree siblings

20T19 x 2, 20T22, 40T26, 40T45, 40T55 x 2

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $80=2^{4} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 80.34 magma: IdentifyGroup(G);
 Character table:  2 4 3 3 4 4 4 2 2 2 2 3 3 3 3 5 1 1 . . 1 . 1 1 1 1 . . . . 1a 2a 2b 2c 2d 2e 10a 10b 10c 5a 4a 4b 4c 4d 2P 1a 1a 1a 1a 1a 1a 5a 5a 5a 5a 2e 2c 2e 2c 3P 1a 2a 2b 2c 2d 2e 10a 10c 10b 5a 4c 4d 4a 4b 5P 1a 2a 2b 2c 2d 2e 2d 2a 2a 1a 4a 4b 4c 4d 7P 1a 2a 2b 2c 2d 2e 10a 10c 10b 5a 4c 4d 4a 4b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 X.3 1 -1 -1 1 1 1 1 -1 -1 1 1 -1 1 -1 X.4 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 X.5 1 -1 1 -1 1 -1 1 -1 -1 1 B -B -B B X.6 1 -1 1 -1 1 -1 1 -1 -1 1 -B B B -B X.7 1 1 -1 -1 1 -1 1 1 1 1 B B -B -B X.8 1 1 -1 -1 1 -1 1 1 1 1 -B -B B B X.9 2 . . -2 -2 2 -2 . . 2 . . . . X.10 2 . . 2 -2 -2 -2 . . 2 . . . . X.11 4 -4 . . 4 . -1 1 1 -1 . . . . X.12 4 4 . . 4 . -1 -1 -1 -1 . . . . X.13 4 . . . -4 . 1 A -A -1 . . . . X.14 4 . . . -4 . 1 -A A -1 . . . . A = -E(5)+E(5)^2+E(5)^3-E(5)^4 = -Sqrt(5) = -r5 B = -E(4) = -Sqrt(-1) = -i 

magma: CharacterTable(G);