Properties

Label 35T7
Order \(140\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_5\times D_7$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $7$
Group :  $D_5\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,4,18,2,16,5,19,3,17)(6,15,9,13,7,11,10,14,8,12)(21,35,24,33,22,31,25,34,23,32)(26,30,29,28,27), (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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$
14:  $D_{7}$
20:  $D_{10}$
28:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $D_{7}$

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,31)( 7,32)( 8,33)( 9,34)(10,35)(11,26)(12,27)(13,28)(14,29)(15,30)(16,21) (17,22)(18,23)(19,24)(20,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $35$ $2$ $( 2, 5)( 3, 4)( 6,31)( 7,35)( 8,34)( 9,33)(10,32)(11,26)(12,30)(13,29)(14,28) (15,27)(16,21)(17,25)(18,24)(19,23)(20,22)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 10, 10, 10, 5 $ $14$ $10$ $( 1, 2, 3, 4, 5)( 6,32, 8,34,10,31, 7,33, 9,35)(11,27,13,29,15,26,12,28,14,30) (16,22,18,24,20,21,17,23,19,25)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
$ 10, 10, 10, 5 $ $14$ $10$ $( 1, 3, 5, 2, 4)( 6,33,10,32, 9,31, 8,35, 7,34)(11,28,15,27,14,26,13,30,12,29) (16,23,20,22,19,21,18,25,17,24)$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
$ 14, 14, 7 $ $10$ $14$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
$ 35 $ $4$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
$ 35 $ $4$ $35$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$
$ 14, 14, 7 $ $10$ $14$ $( 1,11,21,31, 6,16,26)( 2,15,22,35, 7,20,27, 5,12,25,32,10,17,30) ( 3,14,23,34, 8,19,28, 4,13,24,33, 9,18,29)$
$ 35 $ $4$ $35$ $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$
$ 35 $ $4$ $35$ $( 1,13,25,32, 9,16,28, 5,12,24,31, 8,20,27, 4,11,23,35, 7,19,26, 3,15,22,34, 6,18,30, 2,14,21,33,10,17,29)$
$ 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$
$ 14, 14, 7 $ $10$ $14$ $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$
$ 35 $ $4$ $35$ $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$
$ 35 $ $4$ $35$ $( 1,18,35,12,29, 6,23, 5,17,34,11,28,10,22, 4,16,33,15,27, 9,21, 3,20,32,14, 26, 8,25, 2,19,31,13,30, 7,24)$

Group invariants

Order:  $140=2^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [140, 7]
Character table:   
      2  2  2  2  2  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   1
      7  1  .  1  .  1   .  1   .  1   1   1   1  1   1   1   1  1   1   1   1

        1a 2a 2b 2c 5a 10a 5b 10b 7a 14a 35a 35b 7b 14b 35c 35d 7c 14c 35e 35f
     2P 1a 1a 1a 1a 5b  5b 5a  5a 7b  7b 35d 35c 7c  7c 35f 35e 7a  7a 35b 35a
     3P 1a 2a 2b 2c 5b 10b 5a 10a 7c 14c 35f 35e 7a 14a 35b 35a 7b 14b 35d 35c
     5P 1a 2a 2b 2c 1a  2a 1a  2a 7b 14b  7b  7b 7c 14c  7c  7c 7a 14a  7a  7a
     7P 1a 2a 2b 2c 5b 10b 5a 10a 1a  2b  5b  5a 1a  2b  5b  5a 1a  2b  5b  5a
    11P 1a 2a 2b 2c 5a 10a 5b 10b 7c 14c 35e 35f 7a 14a 35a 35b 7b 14b 35c 35d
    13P 1a 2a 2b 2c 5b 10b 5a 10a 7a 14a 35b 35a 7b 14b 35d 35c 7c 14c 35f 35e
    17P 1a 2a 2b 2c 5b 10b 5a 10a 7c 14c 35f 35e 7a 14a 35b 35a 7b 14b 35d 35c
    19P 1a 2a 2b 2c 5a 10a 5b 10b 7b 14b 35c 35d 7c 14c 35e 35f 7a 14a 35a 35b
    23P 1a 2a 2b 2c 5b 10b 5a 10a 7b 14b 35d 35c 7c 14c 35f 35e 7a 14a 35b 35a
    29P 1a 2a 2b 2c 5a 10a 5b 10b 7a 14a 35a 35b 7b 14b 35c 35d 7c 14c 35e 35f
    31P 1a 2a 2b 2c 5a 10a 5b 10b 7c 14c 35e 35f 7a 14a 35a 35b 7b 14b 35c 35d

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

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
C = E(7)^3+E(7)^4
D = E(7)^2+E(7)^5
E = E(7)+E(7)^6
F = 2*E(7)+2*E(7)^6
G = 2*E(7)^2+2*E(7)^5
H = 2*E(7)^3+2*E(7)^4
I = E(35)^9+E(35)^16+E(35)^19+E(35)^26
J = E(35)^4+E(35)^11+E(35)^24+E(35)^31
K = E(35)+E(35)^6+E(35)^29+E(35)^34
L = E(35)^2+E(35)^12+E(35)^23+E(35)^33
M = E(35)^3+E(35)^17+E(35)^18+E(35)^32
N = E(35)^8+E(35)^13+E(35)^22+E(35)^27