Properties

Label 40T14
Order \(40\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $14$
Group :  $C_2\times F_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,3,23)(2,22,4,24)(5,11,40,33)(6,12,39,34)(7,10,37,36)(8,9,38,35)(13,25,29,20)(14,26,30,19)(15,28,31,18)(16,27,32,17), (1,30,33,8)(2,29,34,7)(3,31,36,6)(4,32,35,5)(9,13,28,23)(10,14,27,24)(11,15,25,22)(12,16,26,21)(17,39,20,38)(18,40,19,37)
$|\Aut(F/K)|$:  $40$

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:  $C_4\times C_2$
20:  $F_5$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$ x 3

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

Degree 5: $F_5$

Degree 8: $C_4\times C_2$

Degree 10: $F_5$, $F_{5}\times C_2$ x 2

Degree 20: 20T5, 20T9, 20T13

Low degree siblings

10T5 x 2

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

Group invariants

Order:  $40=2^{3} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [40, 12]
Character table:   
      2  3  3  3  3  3  3  3  3  1   1
      5  1  1  .  .  .  .  .  .  1   1

        1a 2a 2b 2c 4a 4b 4c 4d 5a 10a
     2P 1a 1a 1a 1a 2b 2b 2b 2b 5a  5a
     3P 1a 2a 2b 2c 4c 4d 4a 4b 5a 10a
     5P 1a 2a 2b 2c 4a 4b 4c 4d 1a  2a
     7P 1a 2a 2b 2c 4c 4d 4a 4b 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  1 -1  1 -1  1 -1  1  -1
X.4      1  1  1  1 -1 -1 -1 -1  1   1
X.5      1 -1 -1  1  A -A -A  A  1  -1
X.6      1 -1 -1  1 -A  A  A -A  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