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Properties

Label 40T4
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_5:C_4$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $4$
Group :  $C_2\times C_5:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,18,28,36,3,10,19,25,34)(2,12,17,27,35,4,9,20,26,33)(5,16,24,32,38,7,13,21,29,39)(6,15,23,31,37,8,14,22,30,40), (1,38,4,40)(2,37,3,39)(5,33,8,36)(6,34,7,35)(9,30,11,32)(10,29,12,31)(13,27,15,25)(14,28,16,26)(17,23,19,21)(18,24,20,22)
$|\Aut(F/K)|$:  $40$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
10:  $D_{5}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 5: $D_{5}$

Degree 8: $C_4\times C_2$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T2 x 2, 20T4

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

Group invariants

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

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 -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
X.5      1 -1  1 -1  A -A  A -A  -1   1   1  -1  -1   1   1  -1
X.6      1 -1  1 -1 -A  A -A  A  -1   1   1  -1  -1   1   1  -1
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 -2  2  .  .  .  .   B  -B   B  -B  *B -*B  *B -*B
X.10     2 -2 -2  2  .  .  .  .  *B -*B  *B -*B   B  -B   B  -B
X.11     2 -2  2 -2  .  .  .  .   B  -B  -B   B  *B -*B -*B  *B
X.12     2 -2  2 -2  .  .  .  .  *B -*B -*B  *B   B  -B  -B   B
X.13     2  2 -2 -2  .  .  .  . -*B -*B  *B  *B  -B  -B   B   B
X.14     2  2 -2 -2  .  .  .  .  -B  -B   B   B -*B -*B  *B  *B
X.15     2  2  2  2  .  .  .  . -*B -*B -*B -*B  -B  -B  -B  -B
X.16     2  2  2  2  .  .  .  .  -B  -B  -B  -B -*B -*B -*B -*B

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5