# Properties

 Label 20T39 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 $t$: $39$ Group: $C_2^4:D_5$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $4$ Generators: (1,9,12,20)(2,10,11,19)(3,18,13,8)(4,17,14,7)(5,6)(15,16), (1,3,11,13)(2,4,12,14)(5,20,6,19)(7,18)(8,17)(9,15,10,16)

## 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: $(C_2^4 : C_5) : C_2$ x 3

## Low degree siblings

10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,10)(13,14)(19,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $20$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,19)(14,20)(15,17)(16,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,19)(10,20)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $5$ $2$ $( 3,14)( 4,13)( 5, 6)( 7, 8)( 9,20)(10,19)(15,16)(17,18)$ $4, 4, 4, 4, 2, 2$ $20$ $4$ $( 1, 2)( 3, 9,13,19)( 4,10,14,20)( 5, 8,16,17)( 6, 7,15,18)(11,12)$ $4, 4, 4, 4, 2, 2$ $20$ $4$ $( 1, 3, 2, 4)( 5,19,16,10)( 6,20,15, 9)( 7,17)( 8,18)(11,13,12,14)$ $5, 5, 5, 5$ $32$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$ $4, 4, 4, 4, 2, 2$ $20$ $4$ $( 1, 3,11,13)( 2, 4,12,14)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)$ $5, 5, 5, 5$ $32$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$

## Group invariants

 Order: $160=2^{5} \cdot 5$ Cyclic: no Abelian: no Solvable: yes GAP id: [160, 234]
 Character table:  2 5 5 3 5 5 3 3 . 3 . 5 1 . . . . . . 1 . 1 1a 2a 2b 2c 2d 4a 4b 5a 4c 5b 2P 1a 1a 1a 1a 1a 2c 2d 5b 2a 5a 3P 1a 2a 2b 2c 2d 4a 4b 5b 4c 5a 5P 1a 2a 2b 2c 2d 4a 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 -3 1 -1 1 . 1 . X.8 5 1 -1 1 -3 1 -1 . 1 . X.9 5 1 1 -3 1 1 -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