Properties

Label 15T37
Order \(1500\)
n \(15\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $37$
CHM label :  $1/2[5^{3}:4]S(3)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15), (3,6,9,12,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
20:  $F_5$
60:  $C_{15} : C_4$
300:  $(C_5^2 : C_3):C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: None

Low degree siblings

15T37 x 3, 30T282 x 2, 30T288 x 4

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 $ $1$ $1$ $()$
$ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $5$ $( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 1, 1, 1, 1, 1 $ $12$ $5$ $( 2, 8,14, 5,11)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $4$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 9,15, 6,12)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,15,12, 9, 6)$
$ 5, 5, 5 $ $12$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3,12, 6,15, 9)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $125$ $2$ $( 4,13)( 5,14)( 6,15)( 7,10)( 8,11)( 9,12)$
$ 3, 3, 3, 3, 3 $ $50$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $100$ $15$ $( 1,11, 9, 4,14,12, 7, 2,15,10, 5, 3,13, 8, 6)$
$ 15 $ $100$ $15$ $( 1,11,12, 7, 2, 3,13, 8, 9, 4,14,15,10, 5, 6)$
$ 6, 6, 3 $ $250$ $6$ $( 1, 8, 3, 4, 5, 6)( 2, 9,13,11,15, 7)(10,14,12)$
$ 4, 4, 4, 2, 1 $ $375$ $4$ $( 1, 2, 4, 8)( 5,10)( 6, 9,15,12)( 7,14,13,11)$
$ 4, 4, 4, 2, 1 $ $375$ $4$ $( 1, 2,13, 8)( 4,11,10,14)( 5, 7)( 6,12,15, 9)$

Group invariants

Order:  $1500=2^{2} \cdot 3 \cdot 5^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1500, 115]
Character table:   
      2  2  .  .  .  .  .  .  .  .  .  .  .  2  1   .   .  1  2  2
      3  1  .  .  .  .  .  1  .  .  .  .  .  1  1   1   1  1  .  .
      5  3  3  3  3  3  3  3  3  3  3  3  3  .  1   1   1  .  .  .

        1a 5a 5b 5c 5d 5e 5f 5g 5h 5i 5j 5k 2a 3a 15a 15b 6a 4a 4b
     2P 1a 5a 5b 5e 5d 5c 5f 5g 5h 5i 5k 5j 1a 3a 15a 15b 3a 2a 2a
     3P 1a 5a 5b 5e 5d 5c 5f 5g 5h 5i 5k 5j 2a 1a  5f  5f 2a 4b 4a
     5P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2a 3a  3a  3a 6a 4a 4b
     7P 1a 5a 5b 5e 5d 5c 5f 5g 5h 5i 5k 5j 2a 3a 15b 15a 6a 4b 4a
    11P 1a 5a 5b 5c 5d 5e 5f 5g 5h 5i 5j 5k 2a 3a 15b 15a 6a 4b 4a
    13P 1a 5a 5b 5e 5d 5c 5f 5g 5h 5i 5k 5j 2a 3a 15b 15a 6a 4a 4b

X.1      1  1  1  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   1  1 -1 -1
X.3      1  1  1  1  1  1  1  1  1  1  1  1 -1  1   1   1 -1  C -C
X.4      1  1  1  1  1  1  1  1  1  1  1  1 -1  1   1   1 -1 -C  C
X.5      2  2  2  2  2  2  2  2  2  2  2  2 -2 -1  -1  -1  1  .  .
X.6      2  2  2  2  2  2  2  2  2  2  2  2  2 -1  -1  -1 -1  .  .
X.7      4 -1 -1 -1  4 -1 -1 -1 -1  4 -1 -1  .  4  -1  -1  .  .  .
X.8      4 -1 -1 -1  4 -1 -1 -1 -1  4 -1 -1  . -2   B  /B  .  .  .
X.9      4 -1 -1 -1  4 -1 -1 -1 -1  4 -1 -1  . -2  /B   B  .  .  .
X.10    12  7  2  2  2  2 -3 -3 -3 -3 -3 -3  .  .   .   .  .  .  .
X.11    12  2  2 -3 -3 -3 12  2  2  2 -3 -3  .  .   .   .  .  .  .
X.12    12 -3 -3  2  2  2 12 -3 -3 -3  2  2  .  .   .   .  .  .  .
X.13    12  2 -3 -3  2 -3 -3 -3  7 -3  2  2  .  .   .   .  .  .  .
X.14    12 -3 -3  2  2  2 -3  7  2 -3 -3 -3  .  .   .   .  .  .  .
X.15    12 -3  7 -3  2 -3 -3  2 -3 -3  2  2  .  .   .   .  .  .  .
X.16    12  2 -3  A -3 *A -3  2 -3  2  2  2  .  .   .   .  .  .  .
X.17    12  2 -3 *A -3  A -3  2 -3  2  2  2  .  .   .   .  .  .  .
X.18    12 -3  2  2 -3  2 -3 -3  2  2 *A  A  .  .   .   .  .  .  .
X.19    12 -3  2  2 -3  2 -3 -3  2  2  A *A  .  .   .   .  .  .  .

A = 3*E(5)-2*E(5)^2-2*E(5)^3+3*E(5)^4
  = (-1+5*Sqrt(5))/2 = 2+5b5
B = E(15)^7+E(15)^11+E(15)^13+E(15)^14
  = (1-Sqrt(-15))/2 = -b15
C = -E(4)
  = -Sqrt(-1) = -i