Properties

Label 40T10
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times D_5$

Learn more about

Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $10$
Group :  $C_2^2\times D_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,29)(2,30)(3,32)(4,31)(5,27)(6,28)(7,26)(8,25)(9,24)(10,23)(11,22)(12,21)(13,19)(14,20)(15,17)(16,18)(33,38)(34,37)(35,39)(36,40), (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), (1,18)(2,17)(3,19)(4,20)(5,15)(6,16)(7,14)(8,13)(9,12)(10,11)(21,38)(22,37)(23,40)(24,39)(25,36)(26,35)(27,33)(28,34)(29,32)(30,31)
$|\Aut(F/K)|$:  $40$

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:  $D_{5}$
20:  $D_{10}$ x 3

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 5: $D_{5}$

Degree 8: $C_2^3$

Degree 10: $D_5$, $D_{10}$ x 6

Degree 20: 20T4 x 3, 20T8 x 4

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [40, 13]
Character table:   
      2  3  3  3  3  3  3   2   2   2   2   2   2   2   2  3  3
      5  1  1  .  .  .  .   1   1   1   1   1   1   1   1  1  1

        1a 2a 2b 2c 2d 2e 10a 10b  5a 10c 10d 10e  5b 10f 2f 2g
     2P 1a 1a 1a 1a 1a 1a  5a  5a  5b  5b  5b  5b  5a  5a 1a 1a
     3P 1a 2a 2b 2c 2d 2e 10d 10e  5b 10f 10a 10b  5a 10c 2f 2g
     5P 1a 2a 2b 2c 2d 2e  2g  2f  1a  2a  2g  2f  1a  2a 2f 2g
     7P 1a 2a 2b 2c 2d 2e 10d 10e  5b 10f 10a 10b  5a 10c 2f 2g

X.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
X.3      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
X.5      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
X.7      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
X.9      2 -2  .  .  .  .   A  -A -*A  *A  *A -*A  -A   A  2 -2
X.10     2 -2  .  .  .  .  *A -*A  -A   A   A  -A -*A  *A  2 -2
X.11     2 -2  .  .  .  . -*A  *A  -A   A  -A   A -*A  *A -2  2
X.12     2 -2  .  .  .  .  -A   A -*A  *A -*A  *A  -A   A -2  2
X.13     2  2  .  .  .  .   A   A -*A -*A  *A  *A  -A  -A -2 -2
X.14     2  2  .  .  .  .  *A  *A  -A  -A   A   A -*A -*A -2 -2
X.15     2  2  .  .  .  . -*A -*A  -A  -A  -A  -A -*A -*A  2  2
X.16     2  2  .  .  .  .  -A  -A -*A -*A -*A -*A  -A  -A  2  2

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5