Properties

Label 15T23
Order \(360\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No
Group: $A_5 \times S_3$

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $23$
Group :  $A_5 \times S_3$
CHM label :  $A(5)[x]S(3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,7)(4,14)(5,10)(8,13), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
60:  $A_5$
120:  $A_5\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $A_5$

Low degree siblings

18T145, 30T85, 30T94, 30T102, 36T551, 36T552, 36T553, 45T40

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, 2, 1 $ $45$ $2$ $( 2, 3)( 4,10)( 5, 9)( 6,11)( 7,13)( 8,12)(14,15)$
$ 6, 3, 2, 2, 1, 1 $ $60$ $6$ $( 2, 3, 5,12, 8,15)( 6,11)( 7,13,10)( 9,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $15$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$
$ 3, 3, 3, 3, 3 $ $40$ $3$ $( 1, 2, 3)( 4,14, 9)( 5,15,10)( 6, 7, 8)(11,12,13)$
$ 15 $ $24$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
$ 15 $ $24$ $15$ $( 1, 2, 3,10,14, 6, 7, 8,15, 4,11,12,13, 5, 9)$
$ 10, 5 $ $36$ $10$ $( 1, 2, 4, 5,13,11, 7,14,10, 8)( 3, 6,12, 9,15)$
$ 10, 5 $ $36$ $10$ $( 1, 2, 4, 8,10,11, 7,14,13, 5)( 3,15, 6,12, 9)$
$ 6, 6, 3 $ $30$ $6$ $( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$
$ 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 $ $2$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$

Group invariants

Order:  $360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [360, 121]
Character table:   
      2  3  1  3  1  3  3  .   .   .   1   1  2  1  1  2
      3  2  2  .  1  1  1  2   1   1   .   .  1  1  1  2
      5  1  .  .  .  .  1  .   1   1   1   1  .  1  1  1

        1a 3a 2a 6a 2b 2c 3b 15a 15b 10a 10b 6b 5a 5b 3c
     2P 1a 3a 1a 3a 1a 1a 3b 15b 15a  5a  5b 3c 5b 5a 3c
     3P 1a 1a 2a 2c 2b 2c 1a  5a  5b 10b 10a 2b 5b 5a 1a
     5P 1a 3a 2a 6a 2b 2c 3b  3c  3c  2c  2c 6b 1a 1a 3c
     7P 1a 3a 2a 6a 2b 2c 3b 15b 15a 10b 10a 6b 5b 5a 3c
    11P 1a 3a 2a 6a 2b 2c 3b 15a 15b 10a 10b 6b 5a 5b 3c
    13P 1a 3a 2a 6a 2b 2c 3b 15b 15a 10b 10a 6b 5b 5a 3c

X.1      1  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  1  1
X.3      2  2  .  .  2  . -1  -1  -1   .   . -1  2  2 -1
X.4      3  .  1  . -1 -3  .   A  *A  -A -*A -1 *A  A  3
X.5      3  .  1  . -1 -3  .  *A   A -*A  -A -1  A *A  3
X.6      3  . -1  . -1  3  .   A  *A   A  *A -1 *A  A  3
X.7      3  . -1  . -1  3  .  *A   A  *A   A -1  A *A  3
X.8      4  1  . -1  . -4  1  -1  -1   1   1  . -1 -1  4
X.9      4  1  .  1  .  4  1  -1  -1  -1  -1  . -1 -1  4
X.10     5 -1 -1  1  1 -5 -1   .   .   .   .  1  .  .  5
X.11     5 -1  1 -1  1  5 -1   .   .   .   .  1  .  .  5
X.12     6  .  .  . -2  .  .  -A -*A   .   .  1  B *B -3
X.13     6  .  .  . -2  .  . -*A  -A   .   .  1 *B  B -3
X.14     8  2  .  .  .  . -1   1   1   .   .  . -2 -2 -4
X.15    10 -2  .  .  2  .  1   .   .   .   . -1  .  . -5

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