↑

Properties

Label 20T20
Order \(80\)
n \(20\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times F_5$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $20$
Group :  $C_4\times F_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,2,11)(3,5,19,18)(4,6,20,17)(7,14,15,9)(8,13,16,10), (1,7,6,19)(2,8,5,20)(3,13,4,14)(9,12,17,16)(10,11,18,15), (1,2)(3,8,20,15)(4,7,19,16)(5,13,17,9)(6,14,18,10)(11,12)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 10: $F_{5}\times C_2$

Low degree siblings

20T20 x 3, 40T53 x 2

Siblings are shown with degree $\leq 47$

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 $ $1$ $1$ $()$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $5$ $4$ $( 3, 7,20,16)( 4, 8,19,15)( 5,14,17,10)( 6,13,18, 9)$
$ 4, 4, 4, 4, 1, 1, 1, 1 $ $5$ $4$ $( 3,16,20, 7)( 4,15,19, 8)( 5,10,17,14)( 6, 9,18,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 4, 4, 4, 4, 2, 2 $ $5$ $4$ $( 1, 2)( 3, 8,20,15)( 4, 7,19,16)( 5,13,17, 9)( 6,14,18,10)(11,12)$
$ 4, 4, 4, 4, 2, 2 $ $5$ $4$ $( 1, 2)( 3,15,20, 8)( 4,16,19, 7)( 5, 9,17,13)( 6,10,18,14)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3, 2, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$
$ 20 $ $4$ $20$ $( 1, 3, 6, 8,10,11,13,15,17,20, 2, 4, 5, 7, 9,12,14,16,18,19)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3, 9, 8)( 2, 4,10, 7)( 5,16, 6,15)(11,13,19,17)(12,14,20,18)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3,18,15)( 2, 4,17,16)( 5,11,13, 8)( 6,12,14, 7)( 9,19,10,20)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4, 2, 3)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$
$ 20 $ $4$ $20$ $( 1, 4, 6, 7,10,12,13,16,17,19, 2, 3, 5, 8, 9,11,14,15,18,20)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4, 9, 7)( 2, 3,10, 8)( 5,15, 6,16)(11,14,19,18)(12,13,20,17)$
$ 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4,18,16)( 2, 3,17,15)( 5,12,13, 7)( 6,11,14, 8)( 9,20,10,19)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,11,16,20)( 4, 8,12,15,19)$
$ 10, 10 $ $4$ $10$ $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$
$ 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 30]
Character table:    
      2  4  4  4  4  4  4  4  4  4   2  4  4  4   2  4  4  2   2  4  4
      5  1  .  .  .  1  .  .  .  .   1  .  .  .   1  .  .  1   1  1  1

        1a 4a 4b 2a 2b 4c 4d 2c 4e 20a 4f 4g 4h 20b 4i 4j 5a 10a 4k 4l
     2P 1a 2a 2a 1a 1a 2a 2a 1a 2b 10a 2c 2c 2b 10a 2c 2c 5a  5a 2b 2b
     3P 1a 4b 4a 2a 2b 4d 4c 2c 4h 20b 4j 4i 4e 20a 4g 4f 5a 10a 4l 4k
     5P 1a 4a 4b 2a 2b 4c 4d 2c 4e  4k 4f 4g 4h  4l 4i 4j 1a  2b 4k 4l
     7P 1a 4b 4a 2a 2b 4d 4c 2c 4h 20b 4j 4i 4e 20a 4g 4f 5a 10a 4l 4k
    11P 1a 4b 4a 2a 2b 4d 4c 2c 4h 20b 4j 4i 4e 20a 4g 4f 5a 10a 4l 4k
    13P 1a 4a 4b 2a 2b 4c 4d 2c 4e 20a 4f 4g 4h 20b 4i 4j 5a 10a 4k 4l
    17P 1a 4a 4b 2a 2b 4c 4d 2c 4e 20a 4f 4g 4h 20b 4i 4j 5a 10a 4k 4l
    19P 1a 4b 4a 2a 2b 4d 4c 2c 4h 20b 4j 4i 4e 20a 4g 4f 5a 10a 4l 4k

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

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