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Properties

Label 15T13
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 $t$ :  $13$
Group :  $(C_5^2 : C_3):C_2$
CHM label :  $[5^{2}]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,7,4)(2,5,8,11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T14, 25T16, 30T37, 30T38

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, 1, 1, 1, 1, 1 $ $15$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)$
$ 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 $ $50$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$
$ 10, 5 $ $15$ $10$ $( 1, 2, 4, 5, 7, 8,10,11,13,14)( 3,15,12, 9, 6)$
$ 10, 5 $ $15$ $10$ $( 1, 2, 7, 8,13,14, 4, 5,10,11)( 3,12, 6,15, 9)$
$ 10, 5 $ $15$ $10$ $( 1, 2,10,11, 4, 5,13,14, 7, 8)( 3, 9,15, 6,12)$
$ 10, 5 $ $15$ $10$ $( 1, 2,13,14,10,11, 7, 8, 4, 5)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $3$ $5$ $( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $3$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,15,12, 9, 6)$

Group invariants

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

        1a 2a 5a 5b 3a 10a 10b 10c 10d 5c 5d 5e 5f
     2P 1a 1a 5b 5a 3a  5c  5d  5f  5e 5d 5e 5f 5c
     3P 1a 2a 5b 5a 1a 10c 10a 10d 10b 5f 5c 5d 5e
     5P 1a 2a 1a 1a 3a  2a  2a  2a  2a 1a 1a 1a 1a
     7P 1a 2a 5b 5a 3a 10b 10d 10a 10c 5d 5e 5f 5c

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

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