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Properties

Label 20T87
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 $t$ :  $87$
Group :  $C_2\times C_2^4:D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12)(2,11)(3,13)(4,14)(7,8)(17,18), (1,16)(2,15)(3,4)(5,11)(6,12)(7,19)(8,20)(9,17)(10,18), (1,12,2,11)(3,10,13,19)(4,9,14,20)(5,8,6,7)(15,18,16,17)
$|\Aut(F/K)|$:  $2$

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: $(C_2^4 : C_5) : C_2$

Low degree siblings

10T23 x 6, 20T71 x 6, 20T73 x 6, 20T76 x 6, 20T81 x 3, 20T85 x 6, 20T87 x 5, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,15)( 6,16)( 9,20)(10,19)$
$ 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, 1, 1, 1, 1 $ $5$ $2$ $( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,10)(13,14)(19,20)$
$ 4, 4, 4, 4, 2, 1, 1 $ $20$ $4$ $( 3, 9,13,20)( 4,10,14,19)( 5, 7,15,17)( 6, 8,16,18)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $20$ $2$ $( 3, 9)( 4,10)( 5,17)( 6,18)( 7,15)( 8,16)(11,12)(13,20)(14,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $20$ $2$ $( 3,10)( 4, 9)( 5, 7)( 6, 8)(11,12)(13,19)(14,20)(15,17)(16,18)$
$ 4, 4, 4, 4, 2, 1, 1 $ $20$ $4$ $( 3,10,13,19)( 4, 9,14,20)( 5,17,15, 7)( 6,18,16, 8)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,20)(10,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5,16)( 6,15)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,14)( 4,13)( 5,16)( 6,15)( 7,18)( 8,17)( 9,19)(10,20)(11,12)$
$ 4, 4, 4, 4, 4 $ $20$ $4$ $( 1, 3, 2, 4)( 5,10,16,20)( 6, 9,15,19)( 7,17, 8,18)(11,14,12,13)$
$ 4, 4, 4, 4, 4 $ $20$ $4$ $( 1, 3, 2, 4)( 5,19,16, 9)( 6,20,15,10)( 7,17, 8,18)(11,14,12,13)$
$ 10, 10 $ $32$ $10$ $( 1, 3, 5, 8, 9, 2, 4, 6, 7,10)(11,13,15,17,19,12,14,16,18,20)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1, 3, 5,17,19)( 2, 4, 6,18,20)( 7,10,12,14,16)( 8, 9,11,13,15)$
$ 4, 4, 4, 2, 2, 2, 2 $ $20$ $4$ $( 1, 3,12,13)( 2, 4,11,14)( 5,10)( 6, 9)( 7,17, 8,18)(15,19)(16,20)$
$ 4, 4, 4, 2, 2, 2, 2 $ $20$ $4$ $( 1, 3,12,13)( 2, 4,11,14)( 5,19)( 6,20)( 7,17, 8,18)( 9,16)(10,15)$
$ 5, 5, 5, 5 $ $32$ $5$ $( 1, 5, 9,14,17)( 2, 6,10,13,18)( 3, 7,12,16,20)( 4, 8,11,15,19)$
$ 10, 10 $ $32$ $10$ $( 1, 5,19,13,18, 2, 6,20,14,17)( 3, 7,12,16,10, 4, 8,11,15, 9)$

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  4  4  6  6  6  6  4  4   1   1  4  4   1   1
      5  1  .  .  .  .  .  .  .  .  1  .  .  .  .   1   1  .  .   1   1

        1a 2a 2b 2c 4a 2d 2e 4b 2f 2g 2h 2i 4c 4d 10a  5a 4e 4f  5b 10b
     2P 1a 1a 1a 1a 2f 1a 1a 2f 1a 1a 1a 1a 2h 2h  5b  5b 2b 2b  5a  5a
     3P 1a 2a 2b 2c 4a 2d 2e 4b 2f 2g 2h 2i 4c 4d 10b  5b 4e 4f  5a 10a
     5P 1a 2a 2b 2c 4a 2d 2e 4b 2f 2g 2h 2i 4c 4d  2g  1a 4e 4f  1a  2g
     7P 1a 2a 2b 2c 4a 2d 2e 4b 2f 2g 2h 2i 4c 4d 10b  5b 4e 4f  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  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 -2  .  .   A  -A  .  . -*A  *A
X.6      2 -2  2 -2  .  .  .  .  2 -2  2 -2  .  .  *A -*A  .  .  -A   A
X.7      2  2  2  2  .  .  .  .  2  2  2  2  .  . -*A -*A  .  .  -A  -A
X.8      2  2  2  2  .  .  .  .  2  2  2  2  .  .  -A  -A  .  . -*A -*A
X.9      5 -3  1  1 -1  1  1 -1  1  5 -3  1  1  1   .   . -1 -1   .   .
X.10     5 -3  1  1  1 -1 -1  1  1  5 -3  1 -1 -1   .   .  1  1   .   .
X.11     5  3  1 -1 -1 -1  1  1  1 -5 -3 -1 -1  1   .   . -1  1   .   .
X.12     5  3  1 -1  1  1 -1 -1  1 -5 -3 -1  1 -1   .   .  1 -1   .   .
X.13     5 -1 -3  3 -1 -1  1  1  1 -5  1 -1  1 -1   .   .  1 -1   .   .
X.14     5 -1 -3  3  1  1 -1 -1  1 -5  1 -1 -1  1   .   . -1  1   .   .
X.15     5 -1  1 -1 -1  1 -1  1 -3 -5  1  3 -1  1   .   .  1 -1   .   .
X.16     5 -1  1 -1  1 -1  1 -1 -3 -5  1  3  1 -1   .   . -1  1   .   .
X.17     5  1 -3 -3 -1  1  1 -1  1  5  1  1 -1 -1   .   .  1  1   .   .
X.18     5  1 -3 -3  1 -1 -1  1  1  5  1  1  1  1   .   . -1 -1   .   .
X.19     5  1  1  1 -1 -1 -1 -1 -3  5  1 -3  1  1   .   .  1  1   .   .
X.20     5  1  1  1  1  1  1  1 -3  5  1 -3 -1 -1   .   . -1 -1   .   .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5