Properties

Label 15T15
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 $t$ :  $15$
Group :  $\GL(2,4)$
CHM label :  $3A_{5}(15)=[3]A(5)=GL(2,4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,2,15)(4,5,6)(8,9,10)(12,13,14), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13)
$|\Aut(F/K)|$:  $3$

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: None

Degree 5: $A_5$

Low degree siblings

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

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

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

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

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

X.1      1   1  1   1   1   1  1   1  1   1  1   1   1  1  1
X.2      1   A  1  /A  /A   A  1  /A  1   A  1   A  /A /A  A
X.3      1  /A  1   A   A  /A  1   A  1  /A  1  /A   A  A /A
X.4      3   . -1   .   B  -1  B  -1 *B  *B  .   B  *B  3  3
X.5      3   . -1   .  *B  -1 *B  -1  B   B  .  *B   B  3  3
X.6      3   . -1   .   C -/A *B  -A  B   D  .  /C  /D  E /E
X.7      3   . -1   .   D  -A  B -/A *B   C  .  /D  /C /E  E
X.8      3   . -1   .  /D -/A  B  -A *B  /C  .   D   C  E /E
X.9      3   . -1   .  /C  -A *B -/A  B  /D  .   C   D /E  E
X.10     4   1  .   1  -1   . -1   . -1  -1  1  -1  -1  4  4
X.11     4  /A  .   A  -A   . -1   . -1 -/A  1 -/A  -A  F /F
X.12     4   A  .  /A -/A   . -1   . -1  -A  1  -A -/A /F  F
X.13     5  -1  1  -1   .   1  .   1  .   . -1   .   .  5  5
X.14     5 -/A  1  -A   .  /A  .   A  .   . -1   .   .  G /G
X.15     5  -A  1 -/A   .   A  .  /A  .   . -1   .   . /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)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
F = 4*E(3)^2
  = -2-2*Sqrt(-3) = -2-2i3
G = 5*E(3)^2
  = (-5-5*Sqrt(-3))/2 = -5-5b3