Properties

Label 20T38
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 $t$ :  $38$
Group :  $C_2^4:D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $2$

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 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, 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