Properties

Label 35T11
Order \(210\)
n \(35\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{35}:C_3$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
10:  $D_{5}$
30:  $D_5\times C_3$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $F_7$

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

Group invariants

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

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

X.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
X.3      1  A /A -/A  -A -1  1   A  /A  1   A  /A  1   1   1   1   1
X.4      1 /A  A  -A -/A -1  1  /A   A  1  /A   A  1   1   1   1   1
X.5      1  A /A  /A   A  1  1   A  /A  1   A  /A  1   1   1   1   1
X.6      1 /A  A   A  /A  1  1  /A   A  1  /A   A  1   1   1   1   1
X.7      2  B /B   .   .  .  C   E  /E *C   F  /F  2   C  *C  *C   C
X.8      2 /B  B   .   .  .  C  /E   E *C  /F   F  2   C  *C  *C   C
X.9      2  B /B   .   .  . *C   F  /F  C   E  /E  2  *C   C   C  *C
X.10     2 /B  B   .   .  . *C  /F   F  C  /E   E  2  *C   C   C  *C
X.11     2  2  2   .   .  .  C   C   C *C  *C  *C  2   C  *C  *C   C
X.12     2  2  2   .   .  . *C  *C  *C  C   C   C  2  *C   C   C  *C
X.13     6  .  .   .   .  .  6   .   .  6   .   . -1  -1  -1  -1  -1
X.14     6  .  .   .   .  .  D   .   . *D   .   . -1   G   I   J   H
X.15     6  .  .   .   .  .  D   .   . *D   .   . -1   H   J   I   G
X.16     6  .  .   .   .  . *D   .   .  D   .   . -1   I   H   G   J
X.17     6  .  .   .   .  . *D   .   .  D   .   . -1   J   G   H   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)^2+3*E(5)^3
  = (-3-3*Sqrt(5))/2 = -3-3b5
E = E(15)^2+E(15)^8
F = E(15)^11+E(15)^14
G = E(35)^4+E(35)^6+E(35)^9+E(35)^26+E(35)^29+E(35)^31
H = E(35)+E(35)^11+E(35)^16+E(35)^19+E(35)^24+E(35)^34
I = E(35)^8+E(35)^12+E(35)^17+E(35)^18+E(35)^23+E(35)^27
J = E(35)^2+E(35)^3+E(35)^13+E(35)^22+E(35)^32+E(35)^33