Properties

Label 35T30
Order \(5040\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
12:  $C_6\times C_2$
42:  $F_7$
84:  $F_7 \times C_2$
120:  $S_5$
240:  $S_5\times C_2$
360:  $S_5 \times C_3$
720:  30T180

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $F_7$

Low degree siblings

42T417

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 $ $6$ $7$ $( 1,20,35,13,28, 7,24)( 2,19,34,15,30, 6,21)( 3,16,31,14,29, 8,25) ( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,30,34)(20,28,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,19,15,34,21,30)( 7,20,13,35,24,28)( 8,16,14,31,25,29)( 9,17,11,33,23,27) (10,18,12,32,22,26)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,30,21,34,15,19)( 7,28,24,35,13,20)( 8,29,25,31,14,16)( 9,27,23,33,11,17) (10,26,22,32,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$
$ 6, 6, 6, 6, 6, 2, 1, 1, 1 $ $70$ $6$ $( 1, 2)( 6,20,15,35,21,28)( 7,19,13,34,24,30)( 8,16,14,31,25,29) ( 9,17,11,33,23,27)(10,18,12,32,22,26)$
$ 6, 6, 6, 6, 6, 2, 1, 1, 1 $ $70$ $6$ $( 1, 2)( 6,28,21,35,15,20)( 7,30,24,34,13,19)( 8,29,25,31,14,16) ( 9,27,23,33,11,17)(10,26,22,32,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $70$ $2$ $( 1, 2)( 6,35)( 7,34)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,30)(14,29)(15,28) (16,25)(17,23)(18,22)(19,24)(20,21)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1 $ $70$ $6$ $( 1, 2)( 6,13,21, 7,15,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,35,30,20,34,28)$
$ 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 $ $10$ $2$ $( 1, 2)( 6, 7)(13,15)(19,20)(21,24)(28,30)(34,35)$
$ 14, 7, 7, 7 $ $60$ $14$ $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,16,31,14,29, 8,25) ( 4,17,33,11,27, 9,23)( 5,18,32,12,26,10,22)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1 $ $70$ $6$ $( 1, 2)( 6,24,15, 7,21,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,28,34,20,30,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 2)( 4, 5)( 6, 7)( 9,10)(11,12)(13,15)(17,18)(19,20)(21,24)(22,23)(26,27) (28,30)(32,33)(34,35)$
$ 14, 14, 7 $ $90$ $14$ $( 1,19,35,15,28, 6,24, 2,20,34,13,30, 7,21)( 3,16,31,14,29, 8,25) ( 4,18,33,12,27,10,23, 5,17,32,11,26, 9,22)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 4, 5)( 6,24,15, 7,21,13)( 8,25,14)( 9,22,11,10,23,12)(16,29,31) (17,26,33,18,27,32)(19,28,34,20,30,35)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 4, 5)( 6,13,21, 7,15,24)( 8,14,25)( 9,12,23,10,11,22)(16,31,29) (17,32,27,18,33,26)(19,35,30,20,34,28)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 4, 5)( 6,20,15,35,21,28)( 7,19,13,34,24,30)( 8,16,14,31,25,29) ( 9,18,11,32,23,26)(10,17,12,33,22,27)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1, 2)( 4, 5)( 6,28,21,35,15,20)( 7,30,24,34,13,19)( 8,29,25,31,14,16) ( 9,26,23,32,11,18)(10,27,22,33,12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $105$ $2$ $( 1, 2)( 4, 5)( 6,35)( 7,34)( 8,31)( 9,32)(10,33)(11,26)(12,27)(13,30)(14,29) (15,28)(16,25)(17,22)(18,23)(19,24)(20,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1,21,26)( 2,22,28)( 3,25,29)( 4,23,27)( 5,24,30)( 6,32,13)( 7,34,12) ( 8,31,14)( 9,33,11)(10,35,15)(18,20,19)$
$ 21, 7, 7 $ $120$ $21$ $( 1,15,22,35, 6,18,28, 2,12,24,34,10,20,30, 5,13,21,32, 7,19,26) ( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 2, 5)( 6,10, 7)(12,13,15)(18,20,19)(21,22,24)(26,28,30)(32,35,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1, 6,26)( 2,10,28)( 3, 8,29)( 4, 9,27)( 5, 7,30)(12,13,15)(16,31,25) (17,33,23)(18,35,21)(19,32,24)(20,34,22)$
$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $ $140$ $6$ $( 1,30, 5,28, 2,26)( 3,29)( 4,27)( 6,22, 7,21,10,24)( 8,25)( 9,23)(11,17) (12,20,15,18,13,19)(14,16)(32,35,34)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1,34,18, 7,15,26)( 2,32,20, 6,12,28)( 3,31,16, 8,14,29)( 4,33,17, 9,11,27) ( 5,35,19,10,13,30)(21,22,24)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1,19,22,13,34,26)( 2,18,24,15,32,28)( 3,16,25,14,31,29)( 4,17,23,11,33,27) ( 5,20,21,12,35,30)( 6,10, 7)$
$ 6, 6, 6, 6, 6, 3, 2 $ $140$ $6$ $( 1,21,18,28, 6,12)( 2,22,20,30,10,13)( 3,23,16,27, 8,11)( 4,25,17,29, 9,14) ( 5,24,19,26, 7,15)(31,33)(32,35,34)$
$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 2 $ $140$ $6$ $( 1,15, 5,13, 2,12)( 3,11)( 4,14)( 6,10, 7)( 8, 9)(16,33)(17,31) (18,35,19,32,20,34)(21,26,24,30,22,28)(23,29)(25,27)$
$ 6, 6, 6, 6, 6, 3, 2 $ $140$ $6$ $( 1, 6,22,35,30,12)( 2,10,24,34,26,13)( 3, 9,25,33,29,11)( 4, 8,23,31,27,14) ( 5, 7,21,32,28,15)(16,17)(18,20,19)$
$ 21, 14 $ $120$ $42$ $( 1,30,18, 7,34,22,13, 2,26,20, 6,32,24,15, 5,28,19,10,35,21,12) ( 3,27,16, 9,31,23,14, 4,29,17, 8,33,25,11)$
$ 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $ $20$ $6$ $( 1, 2, 5)( 3, 4)( 6,10, 7)( 8, 9)(11,14)(12,13,15)(16,17)(18,20,19)(21,22,24) (23,25)(26,28,30)(27,29)(31,33)(32,35,34)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 3, 2 $ $140$ $6$ $( 1,34,12)( 2,32,13)( 3,33,14, 4,31,11)( 5,35,15)( 6,18,24)( 7,19,22) ( 8,17,25, 9,16,23)(10,20,21)(26,28,30)(27,29)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 3, 2 $ $140$ $6$ $( 1,19,12)( 2,18,13)( 3,17,14, 4,16,11)( 5,20,15)( 6,26,35)( 7,30,32) ( 8,27,31, 9,29,33)(10,28,34)(21,22,24)(23,25)$
$ 12, 12, 6, 4, 1 $ $210$ $12$ $( 1,21,12,17,35, 6, 5,23,13,19,32, 9)( 2,22,11,20,34,10, 4,24,15,18,33, 7) ( 3,25,14,16,31, 8)(26,27,28,30)$
$ 12, 12, 6, 4, 1 $ $210$ $12$ $( 1,15,26,33,24, 6, 5,11,28,34,22, 9)( 2,12,27,35,21,10, 4,13,30,32,23, 7) ( 3,14,29,31,25, 8)(17,20,19,18)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $210$ $4$ $( 1, 6, 5, 9)( 2,10, 4, 7)( 3, 8)(11,35,15,32)(12,33,13,34)(14,31)(16,29) (17,28,19,26)(18,27,20,30)(21,22,23,24)$
$ 12, 12, 4, 3, 3, 1 $ $210$ $12$ $( 1,30,10, 4,28, 6, 5,27, 7, 2,26, 9)( 3,29, 8)(11,13,15,12)(16,25,31) (17,24,34,18,23,35,19,22,33,20,21,32)$
$ 28, 7 $ $180$ $28$ $( 1,34,26,23,20,15,10, 4,35,30,22,17,13, 6, 5,33,28,21,18,11, 7, 2,32,27,24, 19,12, 9)( 3,31,29,25,16,14, 8)$
$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $30$ $4$ $( 1, 2, 5, 4)( 6,10, 9, 7)(11,13,15,12)(17,20,19,18)(21,22,23,24)(26,27,28,30) (32,33,35,34)$
$ 12, 12, 4, 3, 3, 1 $ $210$ $12$ $( 1,19,10, 4,20, 6, 5,17, 7, 2,18, 9)( 3,16, 8)(11,24,30,12,23,28,15,22,27,13, 21,26)(14,25,29)(32,33,35,34)$
$ 15, 15, 5 $ $168$ $15$ $( 1,21,32, 4,25,35, 2,22,33, 3,24,34, 5,23,31)( 6,10, 9, 8, 7)(11,29,20,15,26, 17,14,28,19,12,27,16,13,30,18)$
$ 15, 15, 5 $ $168$ $15$ $( 1,15,32, 4,14,35, 2,12,33, 3,13,34, 5,11,31)( 6,22,17, 8,24,19,10,23,16, 7, 21,18, 9,25,20)(26,27,29,28,30)$
$ 35 $ $144$ $35$ $( 1, 6,12,17,25,28,34, 5, 9,14,20,21,26,33, 3, 7,15,18,23,29,35, 2,10,11,16, 24,30,32, 4, 8,13,19,22,27,31)$
$ 5, 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 2, 5, 4, 3)( 6,10, 9, 8, 7)(11,14,13,15,12)(16,20,19,18,17) (21,22,23,25,24)(26,27,29,28,30)(31,35,34,32,33)$
$ 30, 5 $ $168$ $30$ $( 1,30,12, 9,16,35, 2,26,11, 8,20,34, 5,27,14, 7,19,32, 4,29,13, 6,18,33, 3, 28,15,10,17,31)(21,22,23,25,24)$
$ 10, 10, 10, 5 $ $168$ $10$ $( 1,34, 5,33, 3,35, 2,32, 4,31)( 6,26, 9,29, 7,30,10,27, 8,28)(11,25,13,21,12, 23,14,24,15,22)(16,20,19,18,17)$
$ 30, 5 $ $168$ $30$ $( 1,19,26,23, 8,35, 2,18,27,25, 7,34, 5,17,29,24, 6,32, 4,16,28,21,10,33, 3, 20,30,22, 9,31)(11,14,13,15,12)$

Group invariants

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