Properties

Label 15T16
Order \(180\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $\GL(2,4)$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $16$
Group :  $\GL(2,4)$
CHM label :  $A(5)[x]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13)(2,14)(3,6)(4,7)(8,11)(9,12), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
$|\Aut(F/K)|$:  $3$

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 3: $C_3$

Degree 5: $A_5$

Low degree siblings

15T15 x 2, 18T90, 30T45, 36T176, 45T16

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

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  2  1   2   1   1   1   2   1   1   1  1  1  2  2
      5  1  .  .   .   1   1   .   .   1   1   .  1  1  1  1

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

X.1      1  1  1   1   1   1   1   1   1   1   1  1  1  1  1
X.2      1  1  1   A   A   A   A  /A  /A  /A  /A  1  1 /A  A
X.3      1  1  1  /A  /A  /A  /A   A   A   A   A  1  1  A /A
X.4      3  . -1   .   B  *B  -1   .   B  *B  -1 *B  B  3  3
X.5      3  . -1   .  *B   B  -1   .  *B   B  -1  B *B  3  3
X.6      3  . -1   .   C  /D  -A   .  /C   D -/A  B *B  E /E
X.7      3  . -1   .   D  /C -/A   .  /D   C  -A *B  B /E  E
X.8      3  . -1   .  /D   C  -A   .   D  /C -/A *B  B  E /E
X.9      3  . -1   .  /C   D -/A   .   C  /D  -A  B *B /E  E
X.10     4  1  .   1  -1  -1   .   1  -1  -1   . -1 -1  4  4
X.11     4  1  .  /A -/A -/A   .   A  -A  -A   . -1 -1  F /F
X.12     4  1  .   A  -A  -A   .  /A -/A -/A   . -1 -1 /F  F
X.13     5 -1  1  -1   .   .   1  -1   .   .   1  .  .  5  5
X.14     5 -1  1 -/A   .   .  /A  -A   .   .   A  .  .  G /G
X.15     5 -1  1  -A   .   .   A -/A   .   .  /A  .  . /G  G

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
C = -E(15)-E(15)^4
D = -E(15)^2-E(15)^8
E = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3
F = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
G = 5*E(3)^2
  = (-5-5*Sqrt(-3))/2 = -5-5b3