Properties

Label 20T1
Order \(20\)
n \(20\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{20}$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $1$
Group :  $C_{20}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,12,2,11)(3,14,4,13)(5,15,6,16)(7,17,8,18)(9,20,10,19), (1,4,6,8,9,12,13,16,18,20,2,3,5,7,10,11,14,15,17,19)
$|\Aut(F/K)|$:  $20$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $C_5$

Degree 10: $C_{10}$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 20 $ $1$ $20$ $( 1, 3, 6, 7, 9,11,13,15,18,19, 2, 4, 5, 8,10,12,14,16,17,20)$
$ 20 $ $1$ $20$ $( 1, 4, 6, 8, 9,12,13,16,18,20, 2, 3, 5, 7,10,11,14,15,17,19)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 8,11,16,19)( 4, 7,12,15,20)$
$ 10, 10 $ $1$ $10$ $( 1, 6, 9,13,18, 2, 5,10,14,17)( 3, 7,11,15,19, 4, 8,12,16,20)$
$ 20 $ $1$ $20$ $( 1, 7,13,19, 5,12,17, 3, 9,15, 2, 8,14,20, 6,11,18, 4,10,16)$
$ 20 $ $1$ $20$ $( 1, 8,13,20, 5,11,17, 4, 9,16, 2, 7,14,19, 6,12,18, 3,10,15)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,11,19, 8,16)( 4,12,20, 7,15)$
$ 10, 10 $ $1$ $10$ $( 1,10,18, 6,14, 2, 9,17, 5,13)( 3,12,19, 7,16, 4,11,20, 8,15)$
$ 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 $ $1$ $4$ $( 1,12, 2,11)( 3,14, 4,13)( 5,15, 6,16)( 7,17, 8,18)( 9,20,10,19)$
$ 10, 10 $ $1$ $10$ $( 1,13, 5,17, 9, 2,14, 6,18,10)( 3,15, 8,20,11, 4,16, 7,19,12)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,14, 5,18, 9)( 2,13, 6,17,10)( 3,16, 8,19,11)( 4,15, 7,20,12)$
$ 20 $ $1$ $20$ $( 1,15,10, 3,18,12, 6,19,14, 7, 2,16, 9, 4,17,11, 5,20,13, 8)$
$ 20 $ $1$ $20$ $( 1,16,10, 4,18,11, 6,20,14, 8, 2,15, 9, 3,17,12, 5,19,13, 7)$
$ 10, 10 $ $1$ $10$ $( 1,17,14,10, 5, 2,18,13, 9, 6)( 3,20,16,12, 8, 4,19,15,11, 7)$
$ 5, 5, 5, 5 $ $1$ $5$ $( 1,18,14, 9, 5)( 2,17,13,10, 6)( 3,19,16,11, 8)( 4,20,15,12, 7)$
$ 20 $ $1$ $20$ $( 1,19,17,15,14,11,10, 7, 5, 3, 2,20,18,16,13,12, 9, 8, 6, 4)$
$ 20 $ $1$ $20$ $( 1,20,17,16,14,12,10, 8, 5, 4, 2,19,18,15,13,11, 9, 7, 6, 3)$

Group invariants

Order:  $20=2^{2} \cdot 5$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [20, 2]
Character table:   
      2  2  2   2   2   2   2   2   2   2   2  2  2   2   2   2   2   2   2
      5  1  1   1   1   1   1   1   1   1   1  1  1   1   1   1   1   1   1

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

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

      2   2   2
      5   1   1

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

X.1       1   1
X.2      -1  -1
X.3       A  -A
X.4      -A   A
X.5     -/B  /B
X.6       B  -B
X.7     -/C  /C
X.8       C  -C
X.9      -C   C
X.10     /C -/C
X.11     -B   B
X.12     /B -/B
X.13     /D  /D
X.14     /E  /E
X.15      E   E
X.16      D   D
X.17     -D  -D
X.18     -E  -E
X.19    -/E -/E
X.20    -/D -/D

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(20)
C = -E(20)^13
D = -E(5)
E = -E(5)^2