Properties

Label 35T42
Order \(25200\)
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 $t$ :  $42$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,26,23,31,3,16,28,21,33)(2,17,27,22,32)(4,20,29,25,34,5,19,30,24,35)(6,13)(7,12)(8,11)(9,15)(10,14), (1,29,31,14,6,4,26,34,11,9)(2,28,32,13,7,3,27,33,12,8)(5,30,35,15,10)(16,24)(17,23)(18,22)(19,21)(20,25)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$
5040:  $S_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

Degree 7: $S_7$

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 7, 7, 7, 7, 7 $ $720$ $7$ $( 1, 6,26,31,11,16,21)( 2, 7,27,32,12,17,22)( 3, 8,28,33,13,18,23) ( 4, 9,29,34,14,19,24)( 5,10,30,35,15,20,25)$
$ 35 $ $720$ $35$ $( 1, 9,27,35,13,16,24, 2,10,28,31,14,17,25, 3, 6,29,32,15,18,21, 4, 7,30,33, 11,19,22, 5, 8,26,34,12,20,23)$
$ 35 $ $720$ $35$ $( 1, 7,28,34,15,16,22, 3, 9,30,31,12,18,24, 5, 6,27,33,14,20,21, 2, 8,29,35, 11,17,23, 4,10,26,32,13,19,25)$
$ 35 $ $720$ $35$ $( 1, 8,30,32,14,16,23, 5, 7,29,31,13,20,22, 4, 6,28,35,12,19,21, 3,10,27,34, 11,18,25, 2, 9,26,33,15,17,24)$
$ 35 $ $720$ $35$ $( 1,10,29,33,12,16,25, 4, 8,27,31,15,19,23, 2, 6,30,34,13,17,21, 5, 9,28,32, 11,20,24, 3, 7,26,35,14,18,22)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1008$ $5$ $( 1,10,24,33,27)( 2, 6,25,34,28)( 3, 7,21,35,29)( 4, 8,22,31,30) ( 5, 9,23,32,26)(11,15,14,13,12)(16,20,19,18,17)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1008$ $5$ $( 1, 8,25,32,29)( 2, 9,21,33,30)( 3,10,22,34,26)( 4, 6,23,35,27) ( 5, 7,24,31,28)(11,13,15,12,14)(16,18,20,17,19)$
$ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $504$ $5$ $( 1, 6,21,31,26)( 2, 7,22,32,27)( 3, 8,23,33,28)( 4, 9,24,34,29) ( 5,10,25,35,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 5)( 2, 4)( 6,10)( 7, 9)(11,20)(12,19)(13,18)(14,17)(15,16)(21,25)(22,24) (26,30)(27,29)(31,35)(32,34)$
$ 10, 10, 5, 2, 2, 2, 2, 2 $ $2520$ $10$ $( 1, 7,21,32,26, 2, 6,22,31,27)( 3,10,23,35,28, 5, 8,25,33,30)( 4, 9,24,34,29) (11,17)(12,16)(13,20)(14,19)(15,18)$
$ 15, 15, 5 $ $560$ $15$ $( 1,23,35, 2,24,31, 3,25,32, 4,21,33, 5,22,34)( 6,28,20, 7,29,16, 8,30,17, 9, 26,18,10,27,19)(11,13,15,12,14)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $280$ $3$ $( 1,21,31)( 2,22,32)( 3,23,33)( 4,24,34)( 5,25,35)( 6,26,16)( 7,27,17) ( 8,28,18)( 9,29,19)(10,30,20)$
$ 15, 15, 5 $ $560$ $15$ $( 1,25,34, 3,22,31, 5,24,33, 2,21,35, 4,23,32)( 6,30,19, 8,27,16,10,29,18, 7, 26,20, 9,28,17)(11,15,14,13,12)$
$ 15, 5, 5, 5, 5 $ $140$ $15$ $( 1, 4, 2, 5, 3)( 6,29,17,10,28,16, 9,27,20, 8,26,19, 7,30,18)(11,14,12,15,13) (21,24,22,25,23)(31,34,32,35,33)$
$ 15, 5, 5, 5, 5 $ $140$ $15$ $( 1, 2, 3, 4, 5)( 6,27,18, 9,30,16, 7,28,19,10,26,17, 8,29,20)(11,12,13,14,15) (21,22,23,24,25)(31,32,33,34,35)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 6,26,16)( 7,27,17)( 8,28,18)( 9,29,19)(10,30,20)$
$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $2100$ $6$ $( 2, 5)( 3, 4)( 6,26,16)( 7,30,17,10,27,20)( 8,29,18, 9,28,19)(11,31)(12,35) (13,34)(14,33)(15,32)(22,25)(23,24)$
$ 10, 10, 5, 5, 5 $ $210$ $10$ $( 1, 3, 5, 2, 4)( 6,28,10,27, 9,26, 8,30, 7,29)(11,13,15,12,14) (16,18,20,17,19)(21,33,25,32,24,31,23,35,22,34)$
$ 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 $ $105$ $2$ $( 6,26)( 7,27)( 8,28)( 9,29)(10,30)(21,31)(22,32)(23,33)(24,34)(25,35)$
$ 10, 10, 5, 5, 5 $ $210$ $10$ $( 1, 5, 4, 3, 2)( 6,30, 9,28, 7,26,10,29, 8,27)(11,15,14,13,12) (16,20,19,18,17)(21,35,24,33,22,31,25,34,23,32)$
$ 20, 10, 5 $ $1260$ $20$ $( 1, 2, 3, 4, 5)( 6,32,28,24,10,31,27,23, 9,35,26,22, 8,34,30,21, 7,33,29,25) (11,17,13,19,15,16,12,18,14,20)$
$ 20, 10, 5 $ $1260$ $20$ $( 1, 3, 5, 2, 4)( 6,33,30,22, 9,31,28,25, 7,34,26,23,10,32,29,21, 8,35,27,24) (11,18,15,17,14,16,13,20,12,19)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $630$ $4$ $( 6,31,26,21)( 7,32,27,22)( 8,33,28,23)( 9,34,29,24)(10,35,30,25)(11,16) (12,17)(13,18)(14,19)(15,20)$
$ 15, 10, 10 $ $420$ $30$ $( 1,23, 5,22, 4,21, 3,25, 2,24)( 6,18,30, 7,19,26, 8,20,27, 9,16,28,10,17,29) (11,33,15,32,14,31,13,35,12,34)$
$ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $210$ $6$ $( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,16,26)( 7,17,27)( 8,18,28)( 9,19,29) (10,20,30)(11,31)(12,32)(13,33)(14,34)(15,35)$
$ 15, 10, 10 $ $420$ $30$ $( 1,25, 4,23, 2,21, 5,24, 3,22)( 6,20,29, 8,17,26,10,19,28, 7,16,30, 9,18,27) (11,35,14,33,12,31,15,34,13,32)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $1050$ $4$ $( 1,33,21,13)( 2,32,22,12)( 3,31,23,11)( 4,35,24,15)( 5,34,25,14)( 6, 8) ( 9,10)(16,18)(19,20)(26,28)(29,30)$
$ 6, 6, 4, 4, 4, 4, 4, 3 $ $2100$ $12$ $( 1,13,21,33)( 2,12,22,32)( 3,11,23,31)( 4,15,24,35)( 5,14,25,34) ( 6,28,16, 8,26,18)( 7,27,17)( 9,30,19,10,29,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $525$ $2$ $( 1,30)( 2,29)( 3,28)( 4,27)( 5,26)( 6,25)( 7,24)( 8,23)( 9,22)(10,21)(11,15) (12,14)(16,35)(17,34)(18,33)(19,32)(20,31)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $4200$ $6$ $( 1,18, 6,28,31,23)( 2,17, 7,27,32,22)( 3,16, 8,26,33,21)( 4,20, 9,30,34,25) ( 5,19,10,29,35,24)(11,13)(14,15)$

Group invariants

Order:  $25200=2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.