Properties

Label 40T32
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:D_8$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $32$
Group :  $C_5:D_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (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)(21,24)(22,23), (1,38,4,39,2,37,3,40)(5,34,7,35,6,33,8,36)(9,32,12,29,10,31,11,30)(13,27,16,26,14,28,15,25)(17,23,20,22,18,24,19,21)
$|\Aut(F/K)|$:  $2$

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:  $D_{8}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 15]
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 10a 8a 8b 10b 20a  5a 10c 10d 10e 20b  5b 10f 2c
     2P 1a 1a 1a 2b  5a 4a 4a  5a 10f  5b  5b  5b  5b 10c  5a  5a 1a
     3P 1a 2a 2b 4a 10d 8b 8a 10e 20b  5b 10f 10b 10a 20a  5a 10c 2c
     5P 1a 2a 2b 4a  2c 8b 8a  2c  4a  1a  2b  2c  2c  4a  1a  2b 2c
     7P 1a 2a 2b 4a 10e 8a 8b 10d 20b  5b 10f 10a 10b 20a  5a 10c 2c
    11P 1a 2a 2b 4a 10a 8b 8a 10b 20a  5a 10c 10d 10e 20b  5b 10f 2c
    13P 1a 2a 2b 4a 10d 8b 8a 10e 20b  5b 10f 10b 10a 20a  5a 10c 2c
    17P 1a 2a 2b 4a 10e 8a 8b 10d 20b  5b 10f 10a 10b 20a  5a 10c 2c
    19P 1a 2a 2b 4a 10b 8b 8a 10a 20a  5a 10c 10e 10d 20b  5b 10f 2c

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) = -r2
E = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5