# Properties

 Label 20T38 Degree $20$ Order $160$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^4:D_5$

# Related objects

## Group action invariants

 Degree $n$: $20$ Transitive number $t$: $38$ Group: $C_2^4:D_5$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ Generators: (1,12,2,11)(3,19)(4,20)(5,18,15,7)(6,17,16,8)(9,13)(10,14), (1,10,11,19)(2,9,12,20)(3,8)(4,7)(5,16,6,15)(13,17)(14,18)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $(C_2^4 : C_5) : C_2$

## Low degree siblings

10T15 x 3, 10T16 x 3, 16T415, 20T38 x 5, 20T39, 20T43 x 3, 20T45 x 3, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1$ $5$ $2$ $( 3, 4)( 7,17)( 8,18)( 9,19)(10,20)(13,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $20$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,12)(13,19)(14,20)(15,18)(16,17)$ $4, 4, 4, 4, 2, 1, 1$ $20$ $4$ $( 3,10,14,19)( 4, 9,13,20)( 5, 7,16,17)( 6, 8,15,18)(11,12)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(17,18)$ $4, 4, 4, 4, 4$ $20$ $4$ $( 1, 3, 2, 4)( 5, 9,15,19)( 6,10,16,20)( 7,18, 8,17)(11,13,12,14)$ $5, 5, 5, 5$ $32$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,14,16,18,19)(12,13,15,17,20)$ $4, 4, 4, 2, 2, 2, 2$ $20$ $4$ $( 1, 3,12,14)( 2, 4,11,13)( 5,20)( 6,19)( 7,18, 8,17)( 9,16)(10,15)$ $5, 5, 5, 5$ $32$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,16,19,14,18)(12,15,20,13,17)$

## Group invariants

 Order: $160=2^{5} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [160, 234]
 Character table:  2 5 5 3 3 5 5 3 . 3 . 5 1 . . . . . . 1 . 1 1a 2a 2b 4a 2c 2d 4b 5a 4c 5b 2P 1a 1a 1a 2c 1a 1a 2d 5b 2a 5a 3P 1a 2a 2b 4a 2c 2d 4b 5b 4c 5a 5P 1a 2a 2b 4a 2c 2d 4b 1a 4c 1a 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 2 2 . . 2 2 . A . *A X.4 2 2 . . 2 2 . *A . A X.5 5 -3 -1 1 1 1 1 . -1 . X.6 5 -3 1 -1 1 1 -1 . 1 . X.7 5 1 -1 -1 -3 1 1 . 1 . X.8 5 1 -1 1 1 -3 -1 . 1 . X.9 5 1 1 -1 1 -3 1 . -1 . X.10 5 1 1 1 -3 1 -1 . -1 . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5