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Properties

Label 20T84
Order \(320\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_2\times C_2^4:C_5).C_2$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $84$
Group :  $(C_2\times C_2^4:C_5).C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,12)(3,9)(4,10)(5,15,6,16)(7,13,8,14)(17,20,18,19), (1,7)(2,8)(3,5)(4,6)(9,12,10,11)(13,19)(14,20)(15,17)(16,18)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  20T2
160:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $D_5$

Low degree siblings

20T82 x 6, 20T84 x 2, 40T200 x 3, 40T283 x 3, 40T285 x 6, 40T287 x 6, 40T294 x 6

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

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1583]
Character table:    
      2  6  6  6  6  6  6  6  6  4  4  4  4  4  4  4  4  1   1  1   1
      5  1  .  .  .  .  .  .  1  .  .  .  .  .  .  .  .  1   1  1   1

        1a 2a 2b 2c 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h 5a 10a 5b 10b
     2P 1a 1a 1a 1a 1a 1a 1a 1a 2d 2a 2g 2e 2d 2a 2g 2e 5b  5b 5a  5a
     3P 1a 2a 2b 2c 2d 2e 2f 2g 4e 4f 4g 4h 4a 4b 4c 4d 5b 10b 5a 10a
     5P 1a 2a 2b 2c 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h 1a  2g 1a  2g
     7P 1a 2a 2b 2c 2d 2e 2f 2g 4e 4f 4g 4h 4a 4b 4c 4d 5b 10b 5a 10a

X.1      1  1  1  1  1  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 -1 -1  1   1  1   1
X.3      1 -1  1  1 -1 -1  1 -1  A -A -A  A -A  A  A -A  1  -1  1  -1
X.4      1 -1  1  1 -1 -1  1 -1 -A  A  A -A  A -A -A  A  1  -1  1  -1
X.5      2 -2  2  2 -2 -2  2 -2  .  .  .  .  .  .  .  .  B  -B *B -*B
X.6      2 -2  2  2 -2 -2  2 -2  .  .  .  .  .  .  .  . *B -*B  B  -B
X.7      2  2  2  2  2  2  2  2  .  .  .  .  .  .  .  .  B   B *B  *B
X.8      2  2  2  2  2  2  2  2  .  .  .  .  .  .  .  . *B  *B  B   B
X.9      5 -3  1  1  1  1 -3  5 -1  1  1 -1 -1  1  1 -1  .   .  .   .
X.10     5 -3  1  1  1  1 -3  5  1 -1 -1  1  1 -1 -1  1  .   .  .   .
X.11     5  1 -3  1  1 -3  1  5 -1 -1  1  1 -1 -1  1  1  .   .  .   .
X.12     5  1 -3  1  1 -3  1  5  1  1 -1 -1  1  1 -1 -1  .   .  .   .
X.13     5  1  1 -3 -3  1  1  5 -1  1 -1  1 -1  1 -1  1  .   .  .   .
X.14     5  1  1 -3 -3  1  1  5  1 -1  1 -1  1 -1  1 -1  .   .  .   .
X.15     5  3  1  1 -1 -1 -3 -5  A  A  A  A -A -A -A -A  .   .  .   .
X.16     5  3  1  1 -1 -1 -3 -5 -A -A -A -A  A  A  A  A  .   .  .   .
X.17     5 -1 -3  1 -1  3  1 -5  A -A  A -A -A  A -A  A  .   .  .   .
X.18     5 -1 -3  1 -1  3  1 -5 -A  A -A  A  A -A  A -A  .   .  .   .
X.19     5 -1  1 -3  3 -1  1 -5  A  A -A -A -A -A  A  A  .   .  .   .
X.20     5 -1  1 -3  3 -1  1 -5 -A -A  A  A  A  A -A -A  .   .  .   .

A = -E(4)
  = -Sqrt(-1) = -i
B = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5