Properties

Label 40T12
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{20}$

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Group action invariants

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

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [40, 6]
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 2b 2c 20a 20b  5a 10a 20c 20d 10b  5b 4a
     2P 1a 1a 1a 1a 10a 10a  5b  5b 10b 10b  5a  5a 2a
     3P 1a 2a 2b 2c 20c 20d  5b 10b 20b 20a 10a  5a 4a
     5P 1a 2a 2b 2c  4a  4a  1a  2a  4a  4a  2a  1a 4a
     7P 1a 2a 2b 2c 20d 20c  5b 10b 20a 20b 10a  5a 4a
    11P 1a 2a 2b 2c 20b 20a  5a 10a 20d 20c 10b  5b 4a
    13P 1a 2a 2b 2c 20d 20c  5b 10b 20a 20b 10a  5a 4a
    17P 1a 2a 2b 2c 20c 20d  5b 10b 20b 20a 10a  5a 4a
    19P 1a 2a 2b 2c 20a 20b  5a 10a 20c 20d 10b  5b 4a

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(20)+E(20)^9
B = -E(20)^13+E(20)^17
C = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5