Properties

Label 40T39
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4\times D_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $39$
Group :  $D_4\times D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,3)(5,40)(6,39)(7,37)(8,38)(9,36)(10,35)(11,33)(12,34)(13,32)(14,31)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24), (1,15,25,39,12,23,35,7,20,31,2,16,26,40,11,24,36,8,19,32)(3,13,28,38,10,22,34,5,18,29,4,14,27,37,9,21,33,6,17,30), (1,31)(2,32)(3,29)(4,30)(5,28)(6,27)(7,25)(8,26)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,37)(34,38)(35,39)(36,40)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
10:  $D_{5}$
16:  $D_4\times C_2$
20:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $D_4\times C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8, 20T21 x 2

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, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 5,37)( 6,38)( 7,40)( 8,39)( 9,34)(10,33)(11,35)(12,36)(13,30)(14,29)(15,32) (16,31)(17,28)(18,27)(19,25)(20,26)(21,22)(23,24)$
$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 3)( 2, 4)( 5,39)( 6,40)( 7,38)( 8,37)( 9,35)(10,36)(11,34)(12,33)(13,31) (14,32)(15,29)(16,30)(17,25)(18,26)(19,28)(20,27)(21,24)(22,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 4)( 2, 3)( 5,40)( 6,39)( 7,37)( 8,38)( 9,36)(10,35)(11,33)(12,34)(13,32) (14,31)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,40,10,39)(11,37,12,38)(13,36,14,35)(15,33,16,34) (17,31,18,32)(19,29,20,30)(21,25,22,26)(23,27,24,28)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1, 5,12,14,20,21,26,30,36,38)( 2, 6,11,13,19,22,25,29,35,37)( 3, 7,10,16,18, 24,27,32,33,39)( 4, 8, 9,15,17,23,28,31,34,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,37)(10,38)(11,40)(12,39)(13,34)(14,33)(15,35) (16,36)(17,29)(18,30)(19,31)(20,32)(21,27)(22,28)(23,25)(24,26)$
$ 20, 20 $ $4$ $20$ $( 1, 7,11,15,20,24,25,31,36,39, 2, 8,12,16,19,23,26,32,35,40)( 3, 5, 9,13,18, 21,28,29,33,38, 4, 6,10,14,17,22,27,30,34,37)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1, 9,20,28,36, 4,12,17,26,34)( 2,10,19,27,35, 3,11,18,25,33)( 5,16,21,32,38, 7,14,24,30,39)( 6,15,22,31,37, 8,13,23,29,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,11,20,25,36, 2,12,19,26,35)( 3, 9,18,28,33, 4,10,17,27,34)( 5,13,21,29,38, 6,14,22,30,37)( 7,15,24,31,39, 8,16,23,32,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,20,26,36)( 2,11,19,25,35)( 3,10,18,27,33)( 4, 9,17,28,34) ( 5,14,21,30,38)( 6,13,22,29,37)( 7,16,24,32,39)( 8,15,23,31,40)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,13,26,37,12,22,36, 6,20,29)( 2,14,25,38,11,21,35, 5,19,30)( 3,15,27,40,10, 23,33, 8,18,31)( 4,16,28,39, 9,24,34, 7,17,32)$
$ 20, 20 $ $4$ $20$ $( 1,15,25,39,12,23,35, 7,20,31, 2,16,26,40,11,24,36, 8,19,32)( 3,13,28,38,10, 22,34, 5,18,29, 4,14,27,37, 9,21,33, 6,17,30)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,17,36, 9,26, 4,20,34,12,28)( 2,18,35,10,25, 3,19,33,11,27)( 5,24,38,16,30, 7,21,39,14,32)( 6,23,37,15,29, 8,22,40,13,31)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,36,11,26, 2,20,35,12,25)( 3,17,33, 9,27, 4,18,34,10,28)( 5,22,38,13,30, 6,21,37,14,29)( 7,23,39,15,32, 8,24,40,16,31)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,20,36,12,26)( 2,19,35,11,25)( 3,18,33,10,27)( 4,17,34, 9,28) ( 5,21,38,14,30)( 6,22,37,13,29)( 7,24,39,16,32)( 8,23,40,15,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,21)( 2,22)( 3,24)( 4,23)( 5,26)( 6,25)( 7,27)( 8,28)( 9,31)(10,32)(11,29) (12,30)(13,35)(14,36)(15,34)(16,33)(17,40)(18,39)(19,37)(20,38)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23, 2,24)( 3,22, 4,21)( 5,27, 6,28)( 7,26, 8,25)( 9,30,10,29)(11,32,12,31) (13,34,14,33)(15,35,16,36)(17,38,18,37)(19,39,20,40)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 39]
Character table:   
      2  4  3  4  3  4  4  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 2d 2e 4a 10a 2f 20a 10b 10c  5a 10d 20b 10e 10f  5b 2g 4b
     2P 1a 1a 1a 1a 1a 1a 2b  5a 1a 10c  5b  5b  5b  5b 10f  5a  5a  5a 1a 2b
     3P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f  5b 10a 20a 10b 10c  5a 2g 4b
     5P 1a 2a 2b 2c 2d 2e 4a  2g 2f  4b  2c  2b  1a  2g  4b  2c  2b  1a 2g 4b
     7P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f  5b 10a 20a 10b 10c  5a 2g 4b
    11P 1a 2a 2b 2c 2d 2e 4a 10a 2f 20a 10b 10c  5a 10d 20b 10e 10f  5b 2g 4b
    13P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f  5b 10a 20a 10b 10c  5a 2g 4b
    17P 1a 2a 2b 2c 2d 2e 4a 10d 2f 20b 10e 10f  5b 10a 20a 10b 10c  5a 2g 4b
    19P 1a 2a 2b 2c 2d 2e 4a 10a 2f 20a 10b 10c  5a 10d 20b 10e 10f  5b 2g 4b

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