Properties

Label 40T25
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_5:OD_{16}$

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Group action invariants

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

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:  $C_4\times C_2$
16:  $C_8:C_2$
20:  $F_5$
40:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 8: $C_8:C_2$

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 33]
Character table:   
      2  4  3  4  3  4  4  3  3  3  3  2   2   2   2
      5  1  1  1  .  .  .  .  .  .  .  1   1   1   1

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

X.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
X.3      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
X.5      1 -1  1  1 -1 -1  B -B -B  B  1  -1  -1   1
X.6      1 -1  1  1 -1 -1 -B  B  B -B  1  -1  -1   1
X.7      1  1  1 -1 -1 -1  B  B -B -B  1   1   1   1
X.8      1  1  1 -1 -1 -1 -B -B  B  B  1   1   1   1
X.9      2  . -2  .  A -A  .  .  .  .  2   .   .  -2
X.10     2  . -2  . -A  A  .  .  .  .  2   .   .  -2
X.11     4 -4  4  .  .  .  .  .  .  . -1   1   1  -1
X.12     4  4  4  .  .  .  .  .  .  . -1  -1  -1  -1
X.13     4  . -4  .  .  .  .  .  .  . -1   C  -C   1
X.14     4  . -4  .  .  .  .  .  .  . -1  -C   C   1

A = -2*E(4)
  = -2*Sqrt(-1) = -2i
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(5)+E(5)^2+E(5)^3-E(5)^4
  = -Sqrt(5) = -r5