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

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.