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Properties

Label 20T85
Order \(320\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^4:D_5$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $85$
Group :  $C_2\times C_2^4:D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,12,3)(2,13,11,4)(5,20,15,10)(6,19,16,9)(7,8)(17,18), (1,5,9,4,18,11,15,19,14,8)(2,6,10,3,17,12,16,20,13,7)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$
20:  $D_{10}$
160:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $(C_2^4 : C_5) : C_2$, $C_2\times (C_2^4 : D_5)$ x 2

Low degree siblings

10T23 x 6, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 3, 20T85 x 5, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 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, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 4)( 5, 6)( 9,19)(10,20)(13,14)(15,16)$
$ 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 $ $20$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,19)(14,20)(15,17)(16,18)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $20$ $4$ $( 3, 9,14,20)( 4,10,13,19)( 5,18, 6,17)( 7,15, 8,16)$
$ 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 $ $20$ $4$ $( 1, 2)( 3, 9, 4,10)( 5,17,15, 7)( 6,18,16, 8)(11,12)(13,19,14,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,18)( 8,17)( 9,20)(10,19)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 3)( 2, 4)( 5,20)( 6,19)( 7,17)( 8,18)( 9,16)(10,15)(11,13)(12,14)$
$ 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 $ $32$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$
$ 10, 10 $ $32$ $10$ $( 1, 3, 5,18,10,11,13,15, 8,20)( 2, 4, 6,17, 9,12,14,16, 7,19)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 3,11,13)( 2, 4,12,14)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)$
$ 4, 4, 4, 4, 2, 2 $ $20$ $4$ $( 1, 3,12,14)( 2, 4,11,13)( 5,19,15, 9)( 6,20,16,10)( 7,18)( 8,17)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$
$ 10, 10 $ $32$ $10$ $( 1, 5,20, 3, 7,11,15,10,13,17)( 2, 6,19, 4, 8,12,16, 9,14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$

Group invariants

Order:  $320=2^{6} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [320, 1636]
Character table:    
      2  6  6  6  6  4  4  6  6  4  4  6  4  4  1   1  4  4  1   1  6
      5  1  .  .  .  .  .  .  .  .  .  .  .  .  1   1  .  .  1   1  1

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

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

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