Properties

Label 15T12
Order \(150\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $(C_5^2 : C_3):C_2$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $12$
Group :  $(C_5^2 : C_3):C_2$
CHM label :  $[5^{2}:2]3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T12, 25T15, 30T35 x 2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $6$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 6, 6, 3 $ $25$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$
$ 3, 3, 3, 3, 3 $ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$
$ 6, 6, 3 $ $25$ $6$ $( 1, 3, 5,13, 6, 2)( 4,15, 8,10, 9,14)( 7,12,11)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$

Group invariants

Order:  $150=2 \cdot 3 \cdot 5^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [150, 6]
Character table:   
      2  1  1  .  .  1   1  1   1  .  .
      3  1  1  .  .  1   1  1   1  .  .
      5  2  .  2  2  .   .  .   .  2  2

        1a 2a 5a 5b 3a  6a 3b  6b 5c 5d
     2P 1a 1a 5b 5a 3b  3b 3a  3a 5d 5c
     3P 1a 2a 5b 5a 1a  2a 1a  2a 5d 5c
     5P 1a 2a 1a 1a 3b  6b 3a  6a 1a 1a

X.1      1  1  1  1  1   1  1   1  1  1
X.2      1 -1  1  1  1  -1  1  -1  1  1
X.3      1 -1  1  1  C  -C /C -/C  1  1
X.4      1 -1  1  1 /C -/C  C  -C  1  1
X.5      1  1  1  1  C   C /C  /C  1  1
X.6      1  1  1  1 /C  /C  C   C  1  1
X.7      6  .  A *A  .   .  .   .  B *B
X.8      6  . *A  A  .   .  .   . *B  B
X.9      6  .  B *B  .   .  .   . *A  A
X.10     6  . *B  B  .   .  .   .  A *A

A = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
B = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4
  = (-3-Sqrt(5))/2 = -2-b5
C = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3