Properties

Label 20T3
Order \(20\)
n \(20\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{10}$

Related objects

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Group action invariants

Degree $n$ :  $20$
Transitive number $t$ :  $3$
Group :  $C_2\times C_{10}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,13,5,17,9,2,14,6,18,10)(3,15,8,20,11,4,16,7,19,12), (1,19,18,16,14,11,9,8,5,3)(2,20,17,15,13,12,10,7,6,4)
$|\Aut(F/K)|$:  $20$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
10:  $C_{10}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 10: $C_{10}$ x 3

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

Group invariants

Order:  $20=2^{2} \cdot 5$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [20, 5]
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 10a 10b  5a 10c 10d 10e  5b 10f 2b 2c 10g  5c 10h 10i 10j  5d
     2P 1a 1a  5a  5a  5b  5b  5c  5c  5d  5d 1a 1a  5a  5a  5b  5b  5c  5c
     3P 1a 2a 10e 10d  5c 10g 10l 10k  5a 10c 2b 2c 10j  5d 10b 10a 10f  5b
     5P 1a 2a  2b  2c  1a  2a  2c  2b  1a  2a 2b 2c  2a  1a  2c  2b  2a  1a
     7P 1a 2a 10i 10h  5b 10f 10b 10a  5d 10j 2b 2c 10c  5a 10l 10k 10g  5c

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

      2   2   2
      5   1   1

        10k 10l
     2P  5d  5d
     3P 10i 10h
     5P  2b  2c
     7P 10e 10d

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

A = -E(5)
B = -E(5)^2