Properties

Label 35T31
Degree $35$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no

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Group action invariants

Degree $n$:  $35$
Transitive number $t$:  $31$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33), (2,5)(3,7)(9,10)(13,14)(16,21)(17,22)(19,26)(27,33)(28,31)(30,32)

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

7T7, 14T46, 21T38, 30T565, 42T411, 42T412, 42T413, 42T418

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5 $ $504$ $5$ $( 1,18,25,30, 2)( 3,15, 4,31,19)( 5,17,27,32,34)( 6, 7,16,28,35) ( 8,21,20, 9,14)(10,29,26,23,13)(11,12,33,24,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 1,31)( 2, 4)( 3,25)( 8,33)( 9,11)(12,14)(15,30)(18,19)(20,22)(21,24)$
$ 10, 10, 5, 5, 5 $ $504$ $10$ $( 1, 3, 2,19,30,31,25, 4,18,15)( 5,27,34,17,32)( 6,16,35, 7,28) ( 8,22,14,24, 9,33,20,12,21,11)(10,26,13,29,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 1,28)( 2,22)( 4,12)( 5,13)( 6, 9)( 7,27)( 8,25)(10,18)(11,26)(14,21)(15,32) (16,17)(19,23)(20,24)(29,31)(30,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1,14,30)( 2,31,15)( 3,33,35)( 5,20,16)( 6,25,18)( 7,11,19)( 8,10, 9) (13,24,17)(21,34,28)(22,29,32)(23,27,26)$
$ 6, 6, 6, 6, 6, 3, 2 $ $840$ $6$ $( 1,34,14,28,30,21)( 2,32,31,22,15,29)( 3,35,33)( 4,12)( 5,17,20,13,16,24) ( 6,10,25, 9,18, 8)( 7,23,11,27,19,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $70$ $3$ $( 4,10,16)( 5,11,24)( 6,14,19)( 7,15, 8)( 9,23,21)(12,17,18)(13,20,26) (25,32,27)(29,31,35)(30,34,33)$
$ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ $420$ $6$ $( 1, 2)( 4,29,10,31,16,35)( 5,24,11)( 6, 8,14, 7,19,15)( 9,21,23) (12,33,17,30,18,34)(13,26,20)(25,27,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 2)( 4,17)( 6, 7)( 8,19)( 9,27)(10,18)(12,16)(14,15)(21,32)(22,28)(23,25) (29,30)(31,34)(33,35)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $210$ $4$ $( 1,28, 2,22)( 4,34,17,31)( 5,20)( 6,23, 7,25)( 8,27,19, 9)(10,33,18,35) (11,26)(12,29,16,30)(13,24)(14,21,15,32)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $210$ $6$ $( 1, 2)( 4,12,10,17,16,18)( 5,24,11)( 6, 8,14, 7,19,15)( 9,32,23,27,21,25) (13,26,20)(22,28)(29,33,31,30,35,34)$
$ 12, 12, 6, 4, 1 $ $420$ $12$ $( 1, 2,28,22)( 4,13,18, 5,16,26,17,24,10,20,12,11)( 6,27,21, 7,19,32,23, 8,14, 25, 9,15)(29,33,31,30,35,34)$
$ 7, 7, 7, 7, 7 $ $720$ $7$ $( 1, 6,10,21,30,20,27)( 2,14,33,31,25,28, 3)( 4,32,22,19,11,29, 7) ( 5,12,26, 9,24,13, 8)(15,34,17,18,35,23,16)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $630$ $4$ $( 1,19,23,32)( 2,20,10,15)( 3,24,17,35)( 5,31, 9,21)( 6,25,12,18)( 7,30,22,26) ( 8,16,34,28)(11,14)(13,33)(27,29)$

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table:   
      2  4  4  1   1  4  3  3  2  .  4  1  1  3  3   2
      3  2  1  .   .  1  .  2  1  .  1  2  1  1  1   1
      5  1  1  1   1  .  .  .  .  .  .  .  .  .  .   .
      7  1  .  .   .  .  .  .  .  1  .  .  .  .  .   .

        1a 2a 5a 10a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a
     2P 1a 1a 5a  5a 1a 2b 3a 3a 7a 1a 3b 3b 2b 3a  6c
     3P 1a 2a 5a 10a 2b 4a 1a 2a 7a 2c 1a 2c 4b 2b  4b
     5P 1a 2a 1a  2a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a
     7P 1a 2a 5a 10a 2b 4a 3a 6a 1a 2c 3b 6b 4b 6c 12a
    11P 1a 2a 5a 10a 2b 4a 3a 6a 7a 2c 3b 6b 4b 6c 12a

X.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
X.3      6  4  1  -1  2  .  3  1 -1  .  .  .  2 -1  -1
X.4      6 -4  1   1  2  .  3 -1 -1  .  .  . -2 -1   1
X.5     14  4 -1  -1  2  . -1  1  .  .  2  . -2 -1   1
X.6     14 -6 -1  -1  2  .  2  .  . -2 -1  1  .  2   .
X.7     14 -4 -1   1  2  . -1 -1  .  .  2  .  2 -1  -1
X.8     14  6 -1   1  2  .  2  .  .  2 -1 -1  .  2   .
X.9     15 -5  .   . -1 -1  3  1  1  3  .  . -1 -1  -1
X.10    15  5  .   . -1 -1  3 -1  1 -3  .  .  1 -1   1
X.11    20  .  .   . -4  .  2  . -1  .  2  .  .  2   .
X.12    21  1  1   1  1 -1 -3  1  . -3  .  . -1  1  -1
X.13    21 -1  1  -1  1 -1 -3 -1  .  3  .  .  1  1   1
X.14    35 -5  .   . -1  1 -1  1  . -1 -1 -1  1 -1   1
X.15    35  5  .   . -1  1 -1 -1  .  1 -1  1 -1 -1  -1