Properties

Label 35T9
Order \(210\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times C_7:C_3$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $9$
Group :  $D_5\times C_7:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,14,3,32,11,5,34,13,2,31,15,4,33,12)(6,20,24,8,17,21,10,19,23,7,16,25,9,18,22)(26,30,29,28,27), (1,12,21,32,6,17,26,2,11,22,31,7,16,27)(3,15,23,35,8,20,28,5,13,25,33,10,18,30)(4,14,24,34,9,19,29)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$
21:  $C_7:C_3$
30:  $D_5\times C_3$
42:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $C_7:C_3$

Low degree siblings

There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $35$ $6$ $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $35$ $6$ $( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 15, 15, 5 $ $14$ $15$ $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
$ 15, 15, 5 $ $14$ $15$ $( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
$ 15, 15, 5 $ $14$ $15$ $( 1, 3, 5, 2, 4)( 6,13,25, 7,14,21, 8,15,22, 9,11,23,10,12,24)(16,33,30,17,34, 26,18,35,27,19,31,28,20,32,29)$
$ 15, 15, 5 $ $14$ $15$ $( 1, 3, 5, 2, 4)( 6,23,15, 7,24,11, 8,25,12, 9,21,13,10,22,14)(16,28,35,17,29, 31,18,30,32,19,26,33,20,27,34)$
$ 7, 7, 7, 7, 7 $ $3$ $7$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
$ 14, 14, 7 $ $15$ $14$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
$ 35 $ $6$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
$ 35 $ $6$ $35$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$
$ 7, 7, 7, 7, 7 $ $3$ $7$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
$ 14, 14, 7 $ $15$ $14$ $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$
$ 35 $ $6$ $35$ $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$
$ 35 $ $6$ $35$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$

Group invariants

Order:  $210=2 \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [210, 2]
Character table:   
      2  1  1  1  1   1   1  .   .   .  .   .   .  1   1   .   .  1   1   .
      3  1  1  1  1   1   1  1   1   1  1   1   1  .   .   .   .  .   .   .
      5  1  1  1  .   .   .  1   1   1  1   1   1  1   .   1   1  1   .   1
      7  1  .  .  1   .   .  1   .   .  1   .   .  1   1   1   1  1   1   1

        1a 3a 3b 2a  6a  6b 5a 15a 15b 5b 15c 15d 7a 14a 35a 35b 7b 14b 35c
     2P 1a 3b 3a 1a  3b  3a 5b 15d 15c 5a 15b 15a 7a  7a 35b 35a 7b  7b 35d
     3P 1a 1a 1a 2a  2a  2a 5b  5b  5b 5a  5a  5a 7b 14b 35d 35c 7a 14a 35b
     5P 1a 3b 3a 2a  6b  6a 1a  3b  3a 1a  3b  3a 7b 14b  7b  7b 7a 14a  7a
     7P 1a 3a 3b 2a  6a  6b 5b 15c 15d 5a 15a 15b 1a  2a  5b  5a 1a  2a  5b
    11P 1a 3b 3a 2a  6b  6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c
    13P 1a 3a 3b 2a  6a  6b 5b 15c 15d 5a 15a 15b 7b 14b 35d 35c 7a 14a 35b
    17P 1a 3b 3a 2a  6b  6a 5b 15d 15c 5a 15b 15a 7b 14b 35d 35c 7a 14a 35b
    19P 1a 3a 3b 2a  6a  6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a
    23P 1a 3b 3a 2a  6b  6a 5b 15d 15c 5a 15b 15a 7a 14a 35b 35a 7b 14b 35d
    29P 1a 3b 3a 2a  6b  6a 5a 15b 15a 5b 15d 15c 7a 14a 35a 35b 7b 14b 35c
    31P 1a 3a 3b 2a  6a  6b 5a 15a 15b 5b 15c 15d 7b 14b 35c 35d 7a 14a 35a

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  A /A -1  -A -/A  1   A  /A  1   A  /A  1  -1   1   1  1  -1   1
X.4      1 /A  A -1 -/A  -A  1  /A   A  1  /A   A  1  -1   1   1  1  -1   1
X.5      1  A /A  1   A  /A  1   A  /A  1   A  /A  1   1   1   1  1   1   1
X.6      1 /A  A  1  /A   A  1  /A   A  1  /A   A  1   1   1   1  1   1   1
X.7      2  B /B  .   .   .  C   E  /E *C   F  /F  2   .   C  *C  2   .   C
X.8      2 /B  B  .   .   .  C  /E   E *C  /F   F  2   .   C  *C  2   .   C
X.9      2  B /B  .   .   . *C   F  /F  C   E  /E  2   .  *C   C  2   .  *C
X.10     2 /B  B  .   .   . *C  /F   F  C  /E   E  2   .  *C   C  2   .  *C
X.11     2  2  2  .   .   .  C   C   C *C  *C  *C  2   .   C  *C  2   .   C
X.12     2  2  2  .   .   . *C  *C  *C  C   C   C  2   .  *C   C  2   .  *C
X.13     3  .  . -3   .   .  3   .   .  3   .   .  G  -G   G   G /G -/G  /G
X.14     3  .  . -3   .   .  3   .   .  3   .   . /G -/G  /G  /G  G  -G   G
X.15     3  .  .  3   .   .  3   .   .  3   .   .  G   G   G   G /G  /G  /G
X.16     3  .  .  3   .   .  3   .   .  3   .   . /G  /G  /G  /G  G   G   G
X.17     6  .  .  .   .   .  D   .   . *D   .   .  H   .   I   J /H   .  /I
X.18     6  .  .  .   .   .  D   .   . *D   .   . /H   .  /I  /J  H   .   I
X.19     6  .  .  .   .   . *D   .   .  D   .   .  H   .   J   I /H   .  /J
X.20     6  .  .  .   .   . *D   .   .  D   .   . /H   .  /J  /I  H   .   J

      2   .
      3   .
      5   1
      7   1

        35d
     2P 35c
     3P 35a
     5P  7a
     7P  5a
    11P 35d
    13P 35a
    17P 35a
    19P 35b
    23P 35c
    29P 35d
    31P 35b

X.1       1
X.2       1
X.3       1
X.4       1
X.5       1
X.6       1
X.7      *C
X.8      *C
X.9       C
X.10      C
X.11     *C
X.12      C
X.13     /G
X.14      G
X.15     /G
X.16      G
X.17     /J
X.18      J
X.19     /I
X.20      I

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3
C = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5
D = 3*E(5)+3*E(5)^4
  = (-3+3*Sqrt(5))/2 = 3b5
E = E(15)^2+E(15)^8
F = E(15)^11+E(15)^14
G = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
H = 2*E(7)^3+2*E(7)^5+2*E(7)^6
  = -1-Sqrt(-7) = -1-i7
I = E(35)^2+E(35)^8+E(35)^18+E(35)^22+E(35)^23+E(35)^32
J = E(35)+E(35)^4+E(35)^9+E(35)^11+E(35)^16+E(35)^29