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Properties

Label 10T40
Degree $10$
Order $7200$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5 \wr C_2$

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Group action invariants

Degree $n$:  $10$
Transitive number $t$:  $40$
Group:  $A_5 \wr C_2$
CHM label:  $[A(5)^{2}]2$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
Generators:  (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10), (2,4,10)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T269, 20T363, 24T9631, 25T88, 30T652, 36T7075, 40T5410

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1 $ $225$ $2$ $( 1, 3)( 2,10)( 5, 7)( 6, 8)$
$ 3, 3, 1, 1, 1, 1 $ $400$ $3$ $( 1, 3, 5)( 6, 8,10)$
$ 5, 5 $ $144$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $144$ $5$ $( 1, 3, 5, 9, 7)( 2, 6, 8,10, 4)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $30$ $2$ $( 2,10)( 6, 8)$
$ 3, 1, 1, 1, 1, 1, 1, 1 $ $40$ $3$ $( 6, 8,10)$
$ 5, 1, 1, 1, 1, 1 $ $24$ $5$ $( 2, 4, 6, 8,10)$
$ 5, 1, 1, 1, 1, 1 $ $24$ $5$ $( 2, 6, 8,10, 4)$
$ 3, 2, 2, 1, 1, 1 $ $600$ $6$ $( 1, 3)( 5, 7)( 6, 8,10)$
$ 5, 2, 2, 1 $ $360$ $10$ $( 1, 3)( 2, 4, 6, 8,10)( 5, 7)$
$ 5, 2, 2, 1 $ $360$ $10$ $( 1, 3)( 2, 6, 8,10, 4)( 5, 7)$
$ 5, 3, 1, 1 $ $480$ $15$ $( 1, 3, 5)( 2, 4, 6, 8,10)$
$ 5, 3, 1, 1 $ $480$ $15$ $( 1, 3, 5)( 2, 6, 8,10, 4)$
$ 5, 5 $ $288$ $5$ $( 1, 3, 5, 7, 9)( 2, 6, 8,10, 4)$
$ 2, 2, 2, 2, 2 $ $60$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 10 $ $720$ $10$ $( 1, 8, 3,10, 5, 2, 7, 4, 9, 6)$
$ 10 $ $720$ $10$ $( 1, 8, 3,10, 5, 4, 9, 2, 7, 6)$
$ 4, 4, 2 $ $900$ $4$ $( 1, 8, 3, 6)( 2, 7,10, 5)( 4, 9)$
$ 6, 2, 2 $ $1200$ $6$ $( 1, 8, 3,10, 5, 6)( 2, 7)( 4, 9)$

Group invariants

Order:  $7200=2^{5} \cdot 3^{2} \cdot 5^{2}$
Cyclic:  no
Abelian:  no
Solvable:  no
Label:  not available
Character table:    
      2  5  5  1   1   1  4  2  2  2  2   2   2   .   .  .  3   1   1  3  1
      3  2  .  2   .   .  1  2  1  1  1   .   .   1   1  .  1   .   .  .  1
      5  2  .  .   2   2  1  1  2  2  .   1   1   1   1  2  1   1   1  .  .

        1a 2a 3a  5a  5b 2b 3b 5c 5d 6a 10a 10b 15a 15b 5e 2c 10c 10d 4a 6b
     2P 1a 1a 3a  5b  5a 1a 3b 5d 5c 3b  5d  5c 15b 15a 5e 1a  5a  5b 2a 3a
     3P 1a 2a 1a  5b  5a 2b 1a 5d 5c 2b 10b 10a  5d  5c 5e 2c 10d 10c 4a 2c
     5P 1a 2a 3a  1a  1a 2b 3b 1a 1a 6a  2b  2b  3b  3b 1a 2c  2c  2c 4a 6b
     7P 1a 2a 3a  5b  5a 2b 3b 5d 5c 6a 10b 10a 15b 15a 5e 2c 10d 10c 4a 6b
    11P 1a 2a 3a  5a  5b 2b 3b 5c 5d 6a 10a 10b 15a 15b 5e 2c 10c 10d 4a 6b
    13P 1a 2a 3a  5b  5a 2b 3b 5d 5c 6a 10b 10a 15b 15a 5e 2c 10d 10c 4a 6b

X.1      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 -1
X.3      6 -2  .   A  *A  2  3  C *C -1   G  *G -*G  -G  1  .   .   .  .  .
X.4      6 -2  .  *A   A  2  3 *C  C -1  *G   G  -G -*G  1  .   .   .  .  .
X.5      8  .  2  -2  -2  4  5  3  3  1  -1  -1   .   . -2  .   .   .  .  .
X.6      9  1  .   B  *B -3  .  D *D  .  *G   G   .   . -1 -3  *G   G  1  .
X.7      9  1  .  *B   B -3  . *D  D  .   G  *G   .   . -1 -3   G  *G  1  .
X.8      9  1  .   B  *B -3  .  D *D  .  *G   G   .   . -1  3 -*G  -G -1  .
X.9      9  1  .  *B   B -3  . *D  D  .   G  *G   .   . -1  3  -G -*G -1  .
X.10    10  2 -2   .   .  6  4  5  5  .   1   1  -1  -1  .  .   .   .  .  .
X.11    16  .  1   1   1  .  4 -4 -4  .   .   .  -1  -1  1 -4   1   1  . -1
X.12    16  .  1   1   1  .  4 -4 -4  .   .   .  -1  -1  1  4  -1  -1  .  1
X.13    18  2  .  -2  -2 -6  .  3  3  .  -1  -1   .   .  3  .   .   .  .  .
X.14    24  .  . -*A  -A -4  3  E *E -1   1   1  -G -*G -1  .   .   .  .  .
X.15    24  .  .  -A -*A -4  3 *E  E -1   1   1 -*G  -G -1  .   .   .  .  .
X.16    25  1  1   .   .  5 -5  .  . -1   .   .   .   .  . -5   .   . -1  1
X.17    25  1  1   .   .  5 -5  .  . -1   .   .   .   .  .  5   .   .  1 -1
X.18    30 -2  .   .   . -2 -3  F *F  1 -*G  -G  *G   G  .  .   .   .  .  .
X.19    30 -2  .   .   . -2 -3 *F  F  1  -G -*G   G  *G  .  .   .   .  .  .
X.20    40  . -2   .   .  4  1 -5 -5  1  -1  -1   1   1  .  .   .   .  .  .

A = -2*E(5)-2*E(5)^4
  = 1-Sqrt(5) = 1-r5
B = -2*E(5)-E(5)^2-E(5)^3-2*E(5)^4
  = (3-Sqrt(5))/2 = 1-b5
C = -4*E(5)-3*E(5)^2-3*E(5)^3-4*E(5)^4
  = (7-Sqrt(5))/2 = 3-b5
D = -3*E(5)-3*E(5)^4
  = (3-3*Sqrt(5))/2 = -3b5
E = 3*E(5)-E(5)^2-E(5)^3+3*E(5)^4
  = -1+2*Sqrt(5) = 1+4b5
F = -5*E(5)-5*E(5)^4
  = (5-5*Sqrt(5))/2 = -5b5
G = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5