Properties

Label 40T35
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_4:D_5$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8:C_2$

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 40]
Character table:   
      2  4  3  4  3  3  4   2  4  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 20a 4c 4d 10a 10b 10c  5a 20b 10d 10e 10f  5b 4e 2d
     2P 1a 1a 1a 1a 2b 2b 10c 2b 2b  5a  5b  5b  5b 10f  5b  5a  5a  5a 2b 1a
     3P 1a 2a 2b 2c 4a 4c 20b 4b 4d 10d 10e 10f  5b 20a 10a 10b 10c  5a 4e 2d
     5P 1a 2a 2b 2c 4a 4b  4e 4c 4d  2d  2c  2b  1a  4e  2d  2c  2b  1a 4e 2d
     7P 1a 2a 2b 2c 4a 4c 20b 4b 4d 10d 10e 10f  5b 20a 10a 10b 10c  5a 4e 2d
    11P 1a 2a 2b 2c 4a 4c 20a 4b 4d 10a 10b 10c  5a 20b 10d 10e 10f  5b 4e 2d
    13P 1a 2a 2b 2c 4a 4b 20b 4c 4d 10d 10e 10f  5b 20a 10a 10b 10c  5a 4e 2d
    17P 1a 2a 2b 2c 4a 4b 20b 4c 4d 10d 10e 10f  5b 20a 10a 10b 10c  5a 4e 2d
    19P 1a 2a 2b 2c 4a 4c 20a 4b 4d 10a 10b 10c  5a 20b 10d 10e 10f  5b 4e 2d

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  .  .  A   . -A  .   .   .  -2   2   .   .   .  -2   2  .  .
X.10     2  . -2  .  . -A   .  A  .   .   .  -2   2   .   .   .  -2   2  .  .
X.11     2  .  2 -2  .  .   B  .  .  -B  *B -*B -*B  *B -*B   B  -B  -B -2  2
X.12     2  .  2 -2  .  .  *B  .  . -*B   B  -B  -B   B  -B  *B -*B -*B -2  2
X.13     2  .  2 -2  .  . -*B  .  .  *B   B  -B  -B  -B   B  *B -*B -*B  2 -2
X.14     2  .  2 -2  .  .  -B  .  .   B  *B -*B -*B -*B  *B   B  -B  -B  2 -2
X.15     2  .  2  2  .  .   B  .  .   B -*B -*B -*B  *B  *B  -B  -B  -B -2 -2
X.16     2  .  2  2  .  .  *B  .  .  *B  -B  -B  -B   B   B -*B -*B -*B -2 -2
X.17     2  .  2  2  .  . -*B  .  . -*B  -B  -B  -B  -B  -B -*B -*B -*B  2  2
X.18     2  .  2  2  .  .  -B  .  .  -B -*B -*B -*B -*B -*B  -B  -B  -B  2  2
X.19     4  . -4  .  .  .   .  .  .   .   .   C  -C   .   .   .  *C -*C  .  .
X.20     4  . -4  .  .  .   .  .  .   .   .  *C -*C   .   .   .   C  -C  .  .

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