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Properties

Label 40T11
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_4$

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Group action invariants

Degree $n$:  $40$
Transitive number $t$:  $11$
Group:  $C_5:D_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $40$
Generators:  (1,14)(2,13)(3,15)(4,16)(5,12)(6,11)(7,9)(8,10)(17,38)(18,37)(19,40)(20,39)(21,35)(22,36)(23,33)(24,34)(25,32)(26,31)(27,30)(28,29), (1,17,2,18)(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)

Low degree resolvents

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

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

Degree 20: 20T4, 20T7, 20T11

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [40, 8]
Character table:    
      2  3  3  2  2   2   2   2   2   2   2   2   2  2
      5  1  1  .  .   1   1   1   1   1   1   1   1  1

        1a 2a 4a 2b 10a 10b  5a 10c 10d 10e 10f  5b 2c
     2P 1a 1a 2a 1a  5a  5a  5b  5b  5b  5b  5a  5a 1a
     3P 1a 2a 4a 2b 10d 10e  5b 10f 10b 10a 10c  5a 2c
     5P 1a 2a 4a 2b  2c  2c  1a  2a  2c  2c  2a  1a 2c
     7P 1a 2a 4a 2b 10e 10d  5b 10f 10a 10b 10c  5a 2c

X.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
X.3      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
X.5      2 -2  .  .   .   .   2  -2   .   .  -2   2  .
X.6      2 -2  .  .   A  -A -*C  *C  -B   B   C  -C  .
X.7      2 -2  .  .   B  -B  -C   C   A  -A  *C -*C  .
X.8      2 -2  .  .  -B   B  -C   C  -A   A  *C -*C  .
X.9      2 -2  .  .  -A   A -*C  *C   B  -B   C  -C  .
X.10     2  2  .  .   C   C -*C -*C  *C  *C  -C  -C -2
X.11     2  2  .  .  *C  *C  -C  -C   C   C -*C -*C -2
X.12     2  2  .  . -*C -*C  -C  -C  -C  -C -*C -*C  2
X.13     2  2  .  .  -C  -C -*C -*C -*C -*C  -C  -C  2

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