Properties

Label 40T13
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_{10}.C_4$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$
20:  $F_5$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 8: $C_8$

Degree 10: $F_5$

Degree 20: 20T5

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

Group invariants

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

        1a 2a 4a 4b  8a  8b  8c  8d 5a 10a
     2P 1a 1a 2a 2a  4a  4a  4b  4b 5a  5a
     3P 1a 2a 4b 4a  8d  8c  8b  8a 5a 10a
     5P 1a 2a 4a 4b  8b  8a  8d  8c 1a  2a
     7P 1a 2a 4b 4a  8c  8d  8a  8b 5a 10a

X.1      1  1  1  1   1   1   1   1  1   1
X.2      1  1  1  1  -1  -1  -1  -1  1   1
X.3      1 -1  A -A   B  -B  /B -/B  1  -1
X.4      1 -1  A -A  -B   B -/B  /B  1  -1
X.5      1 -1 -A  A -/B  /B  -B   B  1  -1
X.6      1 -1 -A  A  /B -/B   B  -B  1  -1
X.7      1  1 -1 -1   A   A  -A  -A  1   1
X.8      1  1 -1 -1  -A  -A   A   A  1   1
X.9      4 -4  .  .   .   .   .   . -1   1
X.10     4  4  .  .   .   .   .   . -1  -1

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)^3