Properties

Label 40T40
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 $t$ :  $40$
Group :  $D_4\times D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32)(2,31)(3,30)(4,29)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,38)(34,37)(35,40)(36,39), (1,35)(2,36)(3,34)(4,33)(5,30)(6,29)(7,32)(8,31)(9,28)(10,27)(11,25)(12,26)(13,21)(14,22)(15,24)(16,23), (1,27,12,34,19,3,26,10,35,17)(2,28,11,33,20,4,25,9,36,18)(5,31,14,39,22,8,30,15,37,24)(6,32,13,40,21,7,29,16,38,23)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ 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: $D_4$

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

Group invariants

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

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