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Properties

Label 20T14
Degree $20$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times A_4$

Related objects

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Group action invariants

Degree $n$:  $20$
Transitive number $t$:  $14$
Group:  $C_5\times A_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $5$
Generators:  (1,8,10,13,20,2,5,12,14,17,4,6,9,16,18)(3,7,11,15,19), (1,12,17,8,13,4,9,20,5,16)(2,11,18,7,14,3,10,19,6,15)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$5$:  $C_5$
$12$:  $A_4$
$15$:  $C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 5: $C_5$

Degree 10: None

Low degree siblings

30T11

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

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [60, 9]
Character table:    
      2  2  .  .  2  2   .   .   2  2   .   .   2  2   .   .   2  2   .   .
      3  1  1  1  .  1   1   1   .  1   1   1   .  1   1   1   .  1   1   1
      5  1  1  1  1  1   1   1   1  1   1   1   1  1   1   1   1  1   1   1

        1a 3a 3b 2a 5a 15a 15b 10a 5b 15c 15d 10b 5c 15e 15f 10c 5d 15g 15h
     2P 1a 3b 3a 1a 5b 15d 15c  5b 5d 15h 15g  5d 5a 15b 15a  5a 5c 15f 15e
     3P 1a 1a 1a 2a 5c  5c  5c 10c 5a  5a  5a 10a 5d  5d  5d 10d 5b  5b  5b
     5P 1a 3b 3a 2a 1a  3b  3a  2a 1a  3b  3a  2a 1a  3b  3a  2a 1a  3b  3a
     7P 1a 3a 3b 2a 5b 15c 15d 10b 5d 15g 15h 10d 5a 15a 15b 10a 5c 15e 15f
    11P 1a 3b 3a 2a 5a 15b 15a 10a 5b 15d 15c 10b 5c 15f 15e 10c 5d 15h 15g
    13P 1a 3a 3b 2a 5c 15e 15f 10c 5a 15a 15b 10a 5d 15g 15h 10d 5b 15c 15d

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

      2   2
      3   .
      5   1

        10d
     2P  5c
     3P 10b
     5P  2a
     7P 10c
    11P 10d
    13P 10b

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(5)^4
C = E(5)^3
D = 3*E(5)^2
E = 3*E(5)
F = E(15)^7
G = E(15)^4
H = E(15)
I = E(15)^13