Properties

Label 40T34
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $Q_8\times D_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $34$
Group :  $Q_8\times D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,2,23)(3,21,4,22)(5,17,6,18)(7,20,8,19)(9,13,10,14)(11,15,12,16)(25,39,26,40)(27,37,28,38)(29,33,30,34)(31,36,32,35), (1,18,2,17)(3,19,4,20)(5,16,6,15)(7,13,8,14)(9,12,10,11)(21,40,22,39)(23,37,24,38)(25,34,26,33)(27,35,28,36)(29,32,30,31), (1,15,26,39,12,23,35,7,19,31,2,16,25,40,11,24,36,8,20,32)(3,14,27,38,9,22,34,6,17,29,4,13,28,37,10,21,33,5,18,30)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$, $Q_8$ x 2
10:  $D_{5}$
20:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8

Low degree siblings

There are no siblings with degree $\leq 10$
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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,37)( 6,38)( 7,39)( 8,40)( 9,33)(10,34)(11,35)(12,36)(13,30)(14,29)(15,31) (16,32)(17,28)(18,27)(19,25)(20,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3, 4)( 5,38)( 6,37)( 7,40)( 8,39)( 9,34)(10,33)(11,36)(12,35)(13,29) (14,30)(15,32)(16,31)(17,27)(18,28)(19,26)(20,25)(21,22)(23,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23) (25,28,26,27)(29,31,30,32)(33,35,34,36)(37,40,38,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,34,12,33)(13,32,14,31) (15,30,16,29)(17,26,18,25)(19,28,20,27)(21,24,22,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,38,12,37)(13,36,14,35)(15,33,16,34) (17,32,18,31)(19,29,20,30)(21,25,22,26)(23,28,24,27)$
$ 20, 20 $ $4$ $20$ $( 1, 5,11,13,19,22,26,30,36,37, 2, 6,12,14,20,21,25,29,35,38)( 3, 7,10,15,17, 24,27,31,33,39, 4, 8, 9,16,18,23,28,32,34,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,38,10,37)(11,40,12,39)(13,34,14,33)(15,36,16,35) (17,30,18,29)(19,32,20,31)(21,27,22,28)(23,25,24,26)$
$ 20, 20 $ $4$ $20$ $( 1, 7,11,15,19,24,26,31,36,39, 2, 8,12,16,20,23,25,32,35,40)( 3, 6,10,14,17, 21,27,29,33,38, 4, 5, 9,13,18,22,28,30,34,37)$
$ 20, 20 $ $4$ $20$ $( 1, 9,20,27,36, 3,11,18,25,33, 2,10,19,28,35, 4,12,17,26,34)( 5,15,21,32,37, 8,13,24,29,40, 6,16,22,31,38, 7,14,23,30,39)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,11,19,26,36, 2,12,20,25,35)( 3,10,17,27,33, 4, 9,18,28,34)( 5,13,22,30,37, 6,14,21,29,38)( 7,15,24,31,39, 8,16,23,32,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,19,25,36)( 2,11,20,26,35)( 3, 9,17,28,33)( 4,10,18,27,34) ( 5,14,22,29,37)( 6,13,21,30,38)( 7,16,24,32,39)( 8,15,23,31,40)$
$ 20, 20 $ $4$ $20$ $( 1,13,26,37,12,21,35, 5,19,30, 2,14,25,38,11,22,36, 6,20,29)( 3,15,27,39, 9, 23,34, 7,17,31, 4,16,28,40,10,24,33, 8,18,32)$
$ 20, 20 $ $4$ $20$ $( 1,15,26,39,12,23,35, 7,19,31, 2,16,25,40,11,24,36, 8,20,32)( 3,14,27,38, 9, 22,34, 6,17,29, 4,13,28,37,10,21,33, 5,18,30)$
$ 20, 20 $ $4$ $20$ $( 1,17,35,10,25, 3,20,34,12,28, 2,18,36, 9,26, 4,19,33,11,27)( 5,23,38,16,29, 8,21,39,14,31, 6,24,37,15,30, 7,22,40,13,32)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,19,36,12,25)( 2,20,35,11,26)( 3,17,33, 9,28)( 4,18,34,10,27) ( 5,22,37,14,29)( 6,21,38,13,30)( 7,24,39,16,32)( 8,23,40,15,31)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,20,36,11,25, 2,19,35,12,26)( 3,18,33,10,28, 4,17,34, 9,27)( 5,21,37,13,29, 6,22,38,14,30)( 7,23,39,15,32, 8,24,40,16,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,28, 8,27)( 9,31,10,32)(11,29,12,30) (13,35,14,36)(15,34,16,33)(17,40,18,39)(19,38,20,37)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23, 2,24)( 3,22, 4,21)( 5,27, 6,28)( 7,25, 8,26)( 9,29,10,30)(11,32,12,31) (13,33,14,34)(15,35,16,36)(17,37,18,38)(19,40,20,39)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 41]
Character table:   
      2  4  4  4  4  3  3  3   2  3   2   2   3   3   2   2   2   3   3  3  3
      5  1  .  1  .  1  .  .   1  .   1   1   1   1   1   1   1   1   1  1  1

        1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a  5a 20d 20e 20f  5b 10b 4e 4f
     2P 1a 1a 1a 1a 2b 2b 2b 10a 2b 10a 10b  5b  5b 10b 10b 10a  5a  5a 2b 2b
     3P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b  5b 20a 20b 20c  5a 10a 4e 4f
     5P 1a 2a 2b 2c 4a 4b 4c  4e 4d  4f  4a  2b  1a  4e  4f  4a  1a  2b 4e 4f
     7P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b  5b 20a 20b 20c  5a 10a 4e 4f
    11P 1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a  5a 20d 20e 20f  5b 10b 4e 4f
    13P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b  5b 20a 20b 20c  5a 10a 4e 4f
    17P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b  5b 20a 20b 20c  5a 10a 4e 4f
    19P 1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a  5a 20d 20e 20f  5b 10b 4e 4f

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

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
B = -2*E(5)^2-2*E(5)^3
  = 1+Sqrt(5) = 1+r5