Properties

Label 40T21
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 $t$ :  $21$
Group :  $D_4:D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,31,20,7,36,24,12,37,27,16,2,32,19,8,35,23,11,38,28,15)(3,29,17,5,34,22,9,39,25,13,4,30,18,6,33,21,10,40,26,14), (1,15,27,37,11,23,36,7,19,32)(2,16,28,38,12,24,35,8,20,31)(3,14,25,39,10,21,34,5,18,30)(4,13,26,40,9,22,33,6,17,29), (1,30,2,29)(3,31,4,32)(5,28,6,27)(7,25,8,26)(9,23,10,24)(11,21,12,22)(13,19,14,20)(15,18,16,17)(33,37,34,38)(35,40,36,39)
$|\Aut(F/K)|$:  $20$

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: $D_5$, $D_{10}$ x 2

Degree 20: 20T4

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

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  4  4  3  3   2   2   2   3   3   2   2   3   2   3  3  3
      5  1  1  1  .  .  .  .  .   1   1   1   1   1   1   1   1   1   1  1  1

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

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  .  .  .  .  .   B  -B  *B -*B -*B  *B -*B  -B   B  -B  2 -2
X.10     2 -2  2  .  .  .  .  .  *B -*B   B  -B  -B   B  -B -*B  *B -*B  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  .  A -A  .  .   .   .   .   2  -2   .   .   2   .  -2  .  .
X.18     2  . -2  . -A  A  .  .   .   .   .   2  -2   .   .   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