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Properties

Label 18T90
Order \(180\)
n \(18\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $C_3\times A_5$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $90$
Group :  $C_3\times A_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,6)(2,16,4)(3,17,5)(7,10,14)(8,11,15)(9,12,13), (1,15,2,13,3,14)(4,18,5,16,6,17)(7,9,8)(10,12,11)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
60:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $\PSL(2,5)$

Degree 9: None

Low degree siblings

15T15 x 2, 15T16

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

Group invariants

Order:  $180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [180, 19]
Character table:    
      2  2  2  .  .  2   2   .   .  2   2   .   .   .   .  .
      3  2  1  1  1  2   1   1   1  2   1   1   1   2   2  2
      5  1  .  1  1  1   .   1   1  1   .   1   1   .   .  .

        1a 2a 5a 5b 3a  6a 15a 15b 3b  6b 15c 15d  3c  3d 3e
     2P 1a 1a 5b 5a 3b  3b 15d 15c 3a  3a 15b 15a  3d  3c 3e
     3P 1a 2a 5b 5a 1a  2a  5b  5a 1a  2a  5b  5a  1a  1a 1a
     5P 1a 2a 1a 1a 3b  6b  3b  3b 3a  6a  3a  3a  3d  3c 3e
     7P 1a 2a 5b 5a 3a  6a 15b 15a 3b  6b 15d 15c  3c  3d 3e
    11P 1a 2a 5a 5b 3b  6b 15c 15d 3a  6a 15a 15b  3d  3c 3e
    13P 1a 2a 5b 5a 3a  6a 15b 15a 3b  6b 15d 15c  3c  3d 3e

X.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 /B  /B  /B  /B   B  /B  1
X.3      1  1  1  1 /B  /B  /B  /B  B   B   B   B  /B   B  1
X.4      3 -1  A *A  3  -1   A  *A  3  -1   A  *A   .   .  .
X.5      3 -1 *A  A  3  -1  *A   A  3  -1  *A   A   .   .  .
X.6      3 -1  A *A  C  -B   F   G /C -/B  /F  /G   .   .  .
X.7      3 -1  A *A /C -/B  /F  /G  C  -B   F   G   .   .  .
X.8      3 -1 *A  A  C  -B   G   F /C -/B  /G  /F   .   .  .
X.9      3 -1 *A  A /C -/B  /G  /F  C  -B   G   F   .   .  .
X.10     4  . -1 -1  4   .  -1  -1  4   .  -1  -1   1   1  1
X.11     4  . -1 -1  D   .  -B  -B /D   . -/B -/B   B  /B  1
X.12     4  . -1 -1 /D   . -/B -/B  D   .  -B  -B  /B   B  1
X.13     5  1  .  .  5   1   .   .  5   1   .   .  -1  -1 -1
X.14     5  1  .  .  E   B   .   . /E  /B   .   .  -B -/B -1
X.15     5  1  .  . /E  /B   .   .  E   B   .   . -/B  -B -1

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
B = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
C = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
D = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
E = 5*E(3)^2
  = (-5-5*Sqrt(-3))/2 = -5-5b3
F = -E(15)^7-E(15)^13
G = -E(15)-E(15)^4