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Properties

Label 40T44
Order \(80\)
n \(40\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times F_5$

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Group action invariants

Degree $n$ :  $40$
Transitive number $t$ :  $44$
Group :  $C_2^2\times F_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,35,10,25,4,20,34,12,28)(2,18,36,9,26,3,19,33,11,27)(5,22,37,13,31,8,24,40,16,30)(6,21,38,14,32,7,23,39,15,29), (1,21,4,23)(2,22,3,24)(5,36,40,9)(6,35,39,10)(7,34,38,12)(8,33,37,11)(13,18,31,26)(14,17,32,25)(15,20,29,28)(16,19,30,27), (1,5,17,13)(2,6,18,14)(3,7,19,15)(4,8,20,16)(9,29,11,32)(10,30,12,31)(21,26,38,33)(22,25,37,34)(23,27,39,36)(24,28,40,35)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $C_4\times C_2$ x 6, $C_2^3$
20:  $F_5$
40:  $F_{5}\times C_2$ x 3

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: 20T9 x 2, 20T13

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

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 50]
Character table:    
      2  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4   2   2   2  2
      5  1  .  1  .  1  .  1  .  .  .  .  .  .  .  .  .   1   1   1  1

        1a 2a 2b 2c 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h 10a 10b 10c 5a
     2P 1a 1a 1a 1a 1a 1a 1a 1a 2g 2g 2g 2g 2g 2g 2g 2g  5a  5a  5a 5a
     3P 1a 2a 2b 2c 2d 2e 2f 2g 4h 4g 4f 4e 4d 4c 4b 4a 10a 10b 10c 5a
     5P 1a 2a 2b 2c 2d 2e 2f 2g 4a 4b 4c 4d 4e 4f 4g 4h  2d  2f  2b 1a
     7P 1a 2a 2b 2c 2d 2e 2f 2g 4h 4g 4f 4e 4d 4c 4b 4a 10a 10b 10c 5a

X.1      1  1  1  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   1  -1  -1  1
X.3      1 -1 -1  1  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  -1  -1   1  1
X.5      1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  -1  -1   1  1
X.6      1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  -1   1  -1  1
X.7      1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  -1   1  -1  1
X.8      1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1   1   1   1  1
X.9      1 -1 -1  1 -1  1  1 -1  A -A -A  A -A  A  A -A  -1   1  -1  1
X.10     1 -1 -1  1 -1  1  1 -1 -A  A  A -A  A -A -A  A  -1   1  -1  1
X.11     1 -1  1 -1  1 -1  1 -1  A -A  A -A  A -A  A -A   1   1   1  1
X.12     1 -1  1 -1  1 -1  1 -1 -A  A -A  A -A  A -A  A   1   1   1  1
X.13     1  1 -1 -1  1  1 -1 -1  A  A -A -A  A  A -A -A   1  -1  -1  1
X.14     1  1 -1 -1  1  1 -1 -1 -A -A  A  A -A -A  A  A   1  -1  -1  1
X.15     1  1  1  1 -1 -1 -1 -1  A  A  A  A -A -A -A -A  -1  -1   1  1
X.16     1  1  1  1 -1 -1 -1 -1 -A -A -A -A  A  A  A  A  -1  -1   1  1
X.17     4  . -4  . -4  .  4  .  .  .  .  .  .  .  .  .   1  -1   1 -1
X.18     4  . -4  .  4  . -4  .  .  .  .  .  .  .  .  .  -1   1   1 -1
X.19     4  .  4  . -4  . -4  .  .  .  .  .  .  .  .  .   1   1  -1 -1
X.20     4  .  4  .  4  .  4  .  .  .  .  .  .  .  .  .  -1  -1  -1 -1

A = -E(4)
  = -Sqrt(-1) = -i