Properties

Label 35T17
Order \(840\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
7:  $C_7$
14:  $C_{14}$
120:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $C_7$

Low degree siblings

42T138

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 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1,13,23,34, 7,17,29)( 2,12,22,35, 8,16,30)( 3,14,25,32,10,18,26) ( 4,15,21,33, 6,19,28)( 5,11,24,31, 9,20,27)$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1,34,29,23,17,13, 7)( 2,35,30,22,16,12, 8)( 3,32,26,25,18,14,10) ( 4,33,28,21,19,15, 6)( 5,31,27,24,20,11, 9)$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1,23, 7,29,13,34,17)( 2,22, 8,30,12,35,16)( 3,25,10,26,14,32,18) ( 4,21, 6,28,15,33,19)( 5,24, 9,27,11,31,20)$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1,29,17, 7,34,23,13)( 2,30,16, 8,35,22,12)( 3,26,18,10,32,25,14) ( 4,28,19, 6,33,21,15)( 5,27,20, 9,31,24,11)$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1, 7,13,17,23,29,34)( 2, 8,12,16,22,30,35)( 3,10,14,18,25,26,32) ( 4, 6,15,19,21,28,33)( 5, 9,11,20,24,27,31)$
$ 7, 7, 7, 7, 7 $ $1$ $7$ $( 1,17,34,13,29, 7,23)( 2,16,35,12,30, 8,22)( 3,18,32,14,26,10,25) ( 4,19,33,15,28, 6,21)( 5,20,31,11,27, 9,24)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1,33,29,21,17,15, 7, 4,34,28,23,19,13, 6)( 2,35,30,22,16,12, 8) ( 3,32,26,25,18,14,10)( 5,31,27,24,20,11, 9)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1, 6,13,19,23,28,34, 4, 7,15,17,21,29,33)( 2, 8,12,16,22,30,35) ( 3,10,14,18,25,26,32)( 5, 9,11,20,24,27,31)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1,28,17, 6,34,21,13, 4,29,19, 7,33,23,15)( 2,30,16, 8,35,22,12) ( 3,26,18,10,32,25,14)( 5,27,20, 9,31,24,11)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1,19,34,15,29, 6,23, 4,17,33,13,28, 7,21)( 2,16,35,12,30, 8,22) ( 3,18,32,14,26,10,25)( 5,20,31,11,27, 9,24)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1,21, 7,28,13,33,17, 4,23, 6,29,15,34,19)( 2,22, 8,30,12,35,16) ( 3,25,10,26,14,32,18)( 5,24, 9,27,11,31,20)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 4)( 6, 7)(13,15)(17,19)(21,23)(28,29)(33,34)$
$ 14, 7, 7, 7 $ $10$ $14$ $( 1,15,23,33, 7,19,29, 4,13,21,34, 6,17,28)( 2,12,22,35, 8,16,30) ( 3,14,25,32,10,18,26)( 5,11,24,31, 9,20,27)$
$ 14, 14, 7 $ $15$ $14$ $( 1,21, 7,28,13,33,17, 4,23, 6,29,15,34,19)( 2,22, 8,30,12,35,16) ( 3,24,10,27,14,31,18, 5,25, 9,26,11,32,20)$
$ 14, 14, 7 $ $15$ $14$ $( 1,33,29,21,17,15, 7, 4,34,28,23,19,13, 6)( 2,35,30,22,16,12, 8) ( 3,31,26,24,18,11,10, 5,32,27,25,20,14, 9)$
$ 14, 14, 7 $ $15$ $14$ $( 1,19,34,15,29, 6,23, 4,17,33,13,28, 7,21)( 2,16,35,12,30, 8,22) ( 3,20,32,11,26, 9,25, 5,18,31,14,27,10,24)$
$ 14, 14, 7 $ $15$ $14$ $( 1, 6,13,19,23,28,34, 4, 7,15,17,21,29,33)( 2, 8,12,16,22,30,35) ( 3, 9,14,20,25,27,32, 5,10,11,18,24,26,31)$
$ 14, 14, 7 $ $15$ $14$ $( 1,15,23,33, 7,19,29, 4,13,21,34, 6,17,28)( 2,12,22,35, 8,16,30) ( 3,11,25,31,10,20,26, 5,14,24,32, 9,18,27)$
$ 14, 14, 7 $ $15$ $14$ $( 1,28,17, 6,34,21,13, 4,29,19, 7,33,23,15)( 2,30,16, 8,35,22,12) ( 3,27,18, 9,32,24,14, 5,26,20,10,31,25,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 4)( 3, 5)( 6, 7)( 9,10)(11,14)(13,15)(17,19)(18,20)(21,23)(24,25)(26,27) (28,29)(31,32)(33,34)$
$ 21, 7, 7 $ $20$ $21$ $( 1, 6,14,17,21,26,34, 4,10,13,19,25,29,33, 3, 7,15,18,23,28,32) ( 2, 8,12,16,22,30,35)( 5, 9,11,20,24,27,31)$
$ 21, 7, 7 $ $20$ $21$ $( 1,19,32,13,28,10,23, 4,18,34,15,26, 7,21, 3,17,33,14,29, 6,25) ( 2,16,35,12,30, 8,22)( 5,20,31,11,27, 9,24)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 4, 3)( 6,10, 7)(13,15,14)(17,19,18)(21,25,23)(26,29,28)(32,34,33)$
$ 21, 7, 7 $ $20$ $21$ $( 1,28,18, 7,33,25,13, 4,26,17, 6,32,23,15, 3,29,19,10,34,21,14) ( 2,30,16, 8,35,22,12)( 5,27,20, 9,31,24,11)$
$ 21, 7, 7 $ $20$ $21$ $( 1,33,26,23,19,14, 7, 4,32,29,21,18,13, 6, 3,34,28,25,17,15,10) ( 2,35,30,22,16,12, 8)( 5,31,27,24,20,11, 9)$
$ 21, 7, 7 $ $20$ $21$ $( 1,15,25,34, 6,18,29, 4,14,23,33,10,17,28, 3,13,21,32, 7,19,26) ( 2,12,22,35, 8,16,30)( 5,11,24,31, 9,20,27)$
$ 21, 7, 7 $ $20$ $21$ $( 1,21,10,29,15,32,17, 4,25, 7,28,14,34,19, 3,23, 6,26,13,33,18) ( 2,22, 8,30,12,35,16)( 5,24, 9,27,11,31,20)$
$ 21, 14 $ $20$ $42$ $( 1, 6,14,17,21,26,34, 4,10,13,19,25,29,33, 3, 7,15,18,23,28,32) ( 2, 9,12,20,22,27,35, 5, 8,11,16,24,30,31)$
$ 21, 14 $ $20$ $42$ $( 1,19,32,13,28,10,23, 4,18,34,15,26, 7,21, 3,17,33,14,29, 6,25) ( 2,20,35,11,30, 9,22, 5,16,31,12,27, 8,24)$
$ 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $ $20$ $6$ $( 1, 4, 3)( 2, 5)( 6,10, 7)( 8, 9)(11,12)(13,15,14)(16,20)(17,19,18)(21,25,23) (22,24)(26,29,28)(27,30)(31,35)(32,34,33)$
$ 21, 14 $ $20$ $42$ $( 1,28,18, 7,33,25,13, 4,26,17, 6,32,23,15, 3,29,19,10,34,21,14) ( 2,27,16, 9,35,24,12, 5,30,20, 8,31,22,11)$
$ 21, 14 $ $20$ $42$ $( 1,33,26,23,19,14, 7, 4,32,29,21,18,13, 6, 3,34,28,25,17,15,10) ( 2,31,30,24,16,11, 8, 5,35,27,22,20,12, 9)$
$ 21, 14 $ $20$ $42$ $( 1,15,25,34, 6,18,29, 4,14,23,33,10,17,28, 3,13,21,32, 7,19,26) ( 2,11,22,31, 8,20,30, 5,12,24,35, 9,16,27)$
$ 21, 14 $ $20$ $42$ $( 1,21,10,29,15,32,17, 4,25, 7,28,14,34,19, 3,23, 6,26,13,33,18) ( 2,24, 8,27,12,31,16, 5,22, 9,30,11,35,20)$
$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $30$ $4$ $( 1, 4, 3, 5)( 6,10, 9, 7)(11,13,15,14)(17,19,18,20)(21,25,24,23)(26,27,29,28) (31,34,33,32)$
$ 28, 7 $ $30$ $28$ $( 1,15,25,31, 7,19,26, 5,13,21,32, 9,17,28, 3,11,23,33,10,20,29, 4,14,24,34, 6,18,27)( 2,12,22,35, 8,16,30)$
$ 28, 7 $ $30$ $28$ $( 1,33,26,24,17,15,10, 5,34,28,25,20,13, 6, 3,31,29,21,18,11, 7, 4,32,27,23, 19,14, 9)( 2,35,30,22,16,12, 8)$
$ 28, 7 $ $30$ $28$ $( 1,21,10,27,13,33,18, 5,23, 6,26,11,34,19, 3,24, 7,28,14,31,17, 4,25, 9,29, 15,32,20)( 2,22, 8,30,12,35,16)$
$ 28, 7 $ $30$ $28$ $( 1,28,18, 9,34,21,14, 5,29,19,10,31,23,15, 3,27,17, 6,32,24,13, 4,26,20, 7, 33,25,11)( 2,30,16, 8,35,22,12)$
$ 28, 7 $ $30$ $28$ $( 1, 6,14,20,23,28,32, 5, 7,15,18,24,29,33, 3, 9,13,19,25,27,34, 4,10,11,17, 21,26,31)( 2, 8,12,16,22,30,35)$
$ 28, 7 $ $30$ $28$ $( 1,19,32,11,29, 6,25, 5,17,33,14,27, 7,21, 3,20,34,15,26, 9,23, 4,18,31,13, 28,10,24)( 2,16,35,12,30, 8,22)$
$ 5, 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 4, 3, 5, 2)( 6,10, 9, 8, 7)(11,12,13,15,14)(16,17,19,18,20) (21,25,24,22,23)(26,27,30,29,28)(31,35,34,33,32)$
$ 35 $ $24$ $35$ $( 1,15,25,31, 8,17,28, 3,11,22,34, 6,18,27, 2,13,21,32, 9,16,29, 4,14,24,35, 7,19,26, 5,12,23,33,10,20,30)$
$ 35 $ $24$ $35$ $( 1,33,26,24,16,13, 6, 3,31,30,23,19,14, 9, 2,34,28,25,20,12, 7, 4,32,27,22, 17,15,10, 5,35,29,21,18,11, 8)$
$ 35 $ $24$ $35$ $( 1,21,10,27,12,34,19, 3,24, 8,29,15,32,20, 2,23, 6,26,11,35,17, 4,25, 9,30, 13,33,18, 5,22, 7,28,14,31,16)$
$ 35 $ $24$ $35$ $( 1,28,18, 9,35,23,15, 3,27,16, 7,33,25,11, 2,29,19,10,31,22,13, 4,26,20, 8, 34,21,14, 5,30,17, 6,32,24,12)$
$ 35 $ $24$ $35$ $( 1, 6,14,20,22,29,33, 3, 9,12,17,21,26,31, 2, 7,15,18,24,30,34, 4,10,11,16, 23,28,32, 5, 8,13,19,25,27,35)$
$ 35 $ $24$ $35$ $( 1,19,32,11,30, 7,21, 3,20,35,13,28,10,24, 2,17,33,14,27, 8,23, 4,18,31,12, 29, 6,25, 5,16,34,15,26, 9,22)$

Group invariants

Order:  $840=2^{3} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [840, 135]
Character table: Data not available.