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Properties

Label 20T39
Order \(160\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4:D_5$

Related objects

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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)
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)
$|\Aut(F/K)|$:  $4$

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 TypeSizeOrderRepresentative
$ 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