Properties

Label 40T3
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:C_8$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $3$
Group :  $C_5:C_8$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,27,10,33,17,3,26,11,35,19,2,28,9,34,18,4,25,12,36,20)(5,31,14,40,23,8,30,15,37,22,6,32,13,39,24,7,29,16,38,21), (1,5,3,8,2,6,4,7)(9,37,12,39,10,38,11,40)(13,34,16,36,14,33,15,35)(17,29,19,31,18,30,20,32)(21,25,23,27,22,26,24,28)
$|\Aut(F/K)|$:  $40$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$
10:  $D_{5}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $D_{5}$

Degree 8: $C_8$

Degree 10: $D_5$

Degree 20: 20T2

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

Group invariants

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

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

X.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
X.3      1 -1  A -A   C  -C  /C -/C  1  -1  -A   A  1  -1   A  -A
X.4      1 -1  A -A  -C   C -/C  /C  1  -1  -A   A  1  -1   A  -A
X.5      1 -1 -A  A -/C  /C  -C   C  1  -1   A  -A  1  -1  -A   A
X.6      1 -1 -A  A  /C -/C   C  -C  1  -1   A  -A  1  -1  -A   A
X.7      1  1 -1 -1   A   A  -A  -A  1   1  -1  -1  1   1  -1  -1
X.8      1  1 -1 -1  -A  -A   A   A  1   1  -1  -1  1   1  -1  -1
X.9      2 -2  B -B   .   .   .   .  D  -D   E  -E *D -*D  -F   F
X.10     2 -2  B -B   .   .   .   . *D -*D   F  -F  D  -D  -E   E
X.11     2 -2 -B  B   .   .   .   .  D  -D  -E   E *D -*D   F  -F
X.12     2 -2 -B  B   .   .   .   . *D -*D  -F   F  D  -D   E  -E
X.13     2  2 -2 -2   .   .   .   .  D   D  -D  -D *D  *D -*D -*D
X.14     2  2 -2 -2   .   .   .   . *D  *D -*D -*D  D   D  -D  -D
X.15     2  2  2  2   .   .   .   .  D   D   D   D *D  *D  *D  *D
X.16     2  2  2  2   .   .   .   . *D  *D  *D  *D  D   D   D   D

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(4)
  = -2*Sqrt(-1) = -2i
C = -E(8)^3
D = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
E = E(20)^13+E(20)^17
F = E(20)+E(20)^9