Properties

Label 40T31
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $Q_8:D_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $31$
Group :  $Q_8:D_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,25,37,12,22,36,5,19,29,2,13,26,38,11,21,35,6,20,30)(3,15,28,40,9,24,33,7,18,31,4,16,27,39,10,23,34,8,17,32), (1,32,3,30,2,31,4,29)(5,25,8,28,6,26,7,27)(9,21,11,24,10,22,12,23)(13,20,15,17,14,19,16,18)(33,38,35,40,34,37,36,39)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$
10:  $D_{5}$
16:  $QD_{16}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $QD_{16}$

Degree 10: $D_{10}$

Degree 20: 20T7

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 17]
Character table:   
      2  4  2  4  3   2  3  3   2   2   3   3   2   2   2   3   3  2
      5  1  .  1  1   1  .  .   1   1   1   1   1   1   1   1   1  1

        1a 2a 2b 4a 20a 8a 8b 20b 20c 10a  5a 20d 20e 20f  5b 10b 4b
     2P 1a 1a 1a 2b 10a 4a 4a 10a 10b  5b  5b 10b 10b 10a  5a  5a 2b
     3P 1a 2a 2b 4a 20d 8a 8b 20e 20f 10b  5b 20b 20a 20c  5a 10a 4b
     5P 1a 2a 2b 4a  4b 8b 8a  4b  4a  2b  1a  4b  4b  4a  1a  2b 4b
     7P 1a 2a 2b 4a 20e 8b 8a 20d 20f 10b  5b 20a 20b 20c  5a 10a 4b
    11P 1a 2a 2b 4a 20a 8a 8b 20b 20c 10a  5a 20d 20e 20f  5b 10b 4b
    13P 1a 2a 2b 4a 20d 8b 8a 20e 20f 10b  5b 20b 20a 20c  5a 10a 4b
    17P 1a 2a 2b 4a 20e 8a 8b 20d 20f 10b  5b 20a 20b 20c  5a 10a 4b
    19P 1a 2a 2b 4a 20b 8a 8b 20a 20c 10a  5a 20e 20d 20f  5b 10b 4b

X.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
X.3      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
X.5      2  .  2 -2   .  .  .   .  -2   2   2   .   .  -2   2   2  .
X.6      2  . -2  .   .  D -D   .   .  -2   2   .   .   .   2  -2  .
X.7      2  . -2  .   . -D  D   .   .  -2   2   .   .   .   2  -2  .
X.8      2  .  2 -2   A  .  .  -A  *C -*C -*C  -B   B   C  -C  -C  .
X.9      2  .  2 -2   B  .  .  -B   C  -C  -C   A  -A  *C -*C -*C  .
X.10     2  .  2 -2  -B  .  .   B   C  -C  -C  -A   A  *C -*C -*C  .
X.11     2  .  2 -2  -A  .  .   A  *C -*C -*C   B  -B   C  -C  -C  .
X.12     2  .  2  2   C  .  .   C -*C -*C -*C  *C  *C  -C  -C  -C -2
X.13     2  .  2  2  *C  .  .  *C  -C  -C  -C   C   C -*C -*C -*C -2
X.14     2  .  2  2 -*C  .  . -*C  -C  -C  -C  -C  -C -*C -*C -*C  2
X.15     2  .  2  2  -C  .  .  -C -*C -*C -*C -*C -*C  -C  -C  -C  2
X.16     4  . -4  .   .  .  .   .   .   E  -E   .   .   . -*E  *E  .
X.17     4  . -4  .   .  .  .   .   .  *E -*E   .   .   .  -E   E  .

A = -E(5)+E(5)^4
B = -E(5)^2+E(5)^3
C = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
D = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
E = -2*E(5)^2-2*E(5)^3
  = 1+Sqrt(5) = 1+r5