Properties

Label 35T4
Order \(70\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{35}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$
14:  $D_{7}$

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

Group invariants

Order:  $70=2 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [70, 3]
Character table:   
      2  1  1  .  .  .   .   .   .   .  .   .   .   .   .  .   .   .   .   .
      5  1  .  1  1  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   1   1

        1a 2a 5a 5b 7a 35a 35b 35c 35d 7b 35e 35f 35g 35h 7c 35i 35j 35k 35l
     2P 1a 1a 5b 5a 7b 35f 35h 35e 35g 7c 35k 35i 35l 35j 7a 35c 35a 35d 35b
     3P 1a 2a 5b 5a 7c 35k 35i 35l 35j 7a 35b 35d 35a 35c 7b 35g 35e 35h 35f
     5P 1a 2a 1a 1a 7b  7b  7b  7b  7b 7c  7c  7c  7c  7c 7a  7a  7a  7a  7a
     7P 1a 2a 5b 5a 1a  5b  5a  5a  5b 1a  5b  5a  5a  5b 1a  5b  5a  5a  5b
    11P 1a 2a 5a 5b 7c 35l 35k 35j 35i 7a 35a 35b 35c 35d 7b 35h 35g 35f 35e
    13P 1a 2a 5b 5a 7a 35b 35d 35a 35c 7b 35f 35h 35e 35g 7c 35j 35l 35i 35k
    17P 1a 2a 5b 5a 7c 35j 35l 35i 35k 7a 35c 35a 35d 35b 7b 35f 35h 35e 35g
    19P 1a 2a 5a 5b 7b 35e 35f 35g 35h 7c 35l 35k 35j 35i 7a 35d 35c 35b 35a
    23P 1a 2a 5b 5a 7b 35g 35e 35h 35f 7c 35j 35l 35i 35k 7a 35b 35d 35a 35c
    29P 1a 2a 5a 5b 7a 35d 35c 35b 35a 7b 35h 35g 35f 35e 7c 35l 35k 35j 35i
    31P 1a 2a 5a 5b 7c 35i 35j 35k 35l 7a 35d 35c 35b 35a 7b 35e 35f 35g 35h

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

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