Properties

Label 40T49
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:SD_{16}$

Learn more about

Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $49$
Group :  $C_5:SD_{16}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,33,3,35,2,34,4,36)(5,31,8,30,6,32,7,29)(9,27,11,25,10,28,12,26)(13,24,15,22,14,23,16,21)(17,40,19,38,18,39,20,37), (1,15,7,17,11,3,13,5,20,10)(2,16,8,18,12,4,14,6,19,9)(21,36,27,39,32)(22,35,28,40,31)(23,33,26,37,30,24,34,25,38,29)
$|\Aut(F/K)|$:  $10$

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: $D_5$

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 16]
Character table:   
      2  4  2  4  3   2   2   3   2   3   2   2   2   3   3  2  3  3
      5  1  1  1  1   1   1   1   1   1   1   1   1   1   1  .  .  .

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

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

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