Properties

Label 35T50
Order \(50400\)
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 $t$ :  $50$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,22,8,16,5,32,23,6,20,2,33,21,10,17,3,31,25,7,18)(4,34,24,9,19)(11,30,12,28)(13,26,15,27)(14,29), (1,8,20,2,9,16,3,10,17,4,6,18,5,7,19)(11,23,15,22,14,21,13,25,12,24)(26,33,30,32,29,31,28,35,27,34)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
20:  $F_5$
5040:  $S_7$
10080:  28T360

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $F_5$

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 $ $4$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ $105$ $4$ $( 1,35, 3,34)( 2,32)( 4,31, 5,33)( 6,10, 8, 9)(11,15,13,14)(16,20,18,19) (21,25,23,24)(26,30,28,29)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 1, 1, 1, 1, 1 $ $105$ $4$ $( 1,32, 5,34)( 2,35, 4,31)( 3,33)( 6, 7,10, 9)(11,12,15,14)(16,17,20,19) (21,22,25,24)(26,27,30,29)$
$ 10, 10, 5, 2, 2, 2, 2, 1, 1 $ $2520$ $10$ $( 2, 5)( 3, 4)( 6,11,21,26,16)( 7,15,22,30,17,10,12,25,27,20)( 8,14,23,29,18, 9,13,24,28,19)(32,35)(33,34)$
$ 5, 5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $504$ $5$ $( 6,11,21,26,16)( 7,12,22,27,17)( 8,13,23,28,18)( 9,14,24,29,19) (10,15,25,30,20)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2016$ $5$ $( 1, 2, 3, 4, 5)( 6,12,23,29,20)( 7,13,24,30,16)( 8,14,25,26,17) ( 9,15,21,27,18)(10,11,22,28,19)(31,32,33,34,35)$
$ 20, 5, 4, 4, 2 $ $2520$ $20$ $( 1,35, 3,34)( 2,32)( 4,31, 5,33)( 6,30,13,19,21,10,28,14,16,25, 8,29,11,20, 23, 9,26,15,18,24)( 7,27,12,17,22)$
$ 20, 5, 4, 4, 2 $ $2520$ $20$ $( 1,32, 5,34)( 2,35, 4,31)( 3,33)( 6,27,15,19,21, 7,30,14,16,22,10,29,11,17, 25, 9,26,12,20,24)( 8,28,13,18,23)$
$ 35 $ $1440$ $35$ $( 1,17,23,14,35,26, 7, 3,19,25,11,32,28, 9, 5,16,22,13,34,30, 6, 2,18,24,15, 31,27, 8, 4,20,21,12,33,29,10)$
$ 35 $ $1440$ $35$ $( 1,18,25,12,34,26, 8, 5,17,24,11,33,30, 7, 4,16,23,15,32,29, 6, 3,20,22,14, 31,28,10, 2,19,21,13,35,27, 9)$
$ 7, 7, 7, 7, 7 $ $720$ $7$ $( 1,16,21,11,31,26, 6)( 2,17,22,12,32,27, 7)( 3,18,23,13,33,28, 8) ( 4,19,24,14,34,29, 9)( 5,20,25,15,35,30,10)$
$ 14, 14, 7 $ $3600$ $14$ $( 1,20,21,15,31,30, 6, 5,16,25,11,35,26,10)( 2,19,22,14,32,29, 7, 4,17,24,12, 34,27, 9)( 3,18,23,13,33,28, 8)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $525$ $4$ $( 1,16)( 2,19, 5,18)( 3,17, 4,20)( 6,26)( 7,29,10,28)( 8,27, 9,30)(11,31) (12,34,15,33)(13,32,14,35)(22,24,25,23)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $525$ $4$ $( 1,16)( 2,18, 5,19)( 3,20, 4,17)( 6,26)( 7,28,10,29)( 8,30, 9,27)(11,31) (12,33,15,34)(13,35,14,32)(22,23,25,24)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $1400$ $6$ $( 1,13,26, 3,11,28)( 2,12,27)( 4,15,29, 5,14,30)( 6,18,31, 8,16,33)( 7,17,32) ( 9,20,34,10,19,35)(21,23)(24,25)$
$ 15, 15, 5 $ $1120$ $15$ $( 1,14,27, 5,13,26, 4,12,30, 3,11,29, 2,15,28)( 6,19,32,10,18,31, 9,17,35, 8, 16,34, 7,20,33)(21,24,22,25,23)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $280$ $3$ $( 1,11,26)( 2,12,27)( 3,13,28)( 4,14,29)( 5,15,30)( 6,16,31)( 7,17,32) ( 8,18,33)( 9,19,34)(10,20,35)$
$ 12, 12, 6, 4, 1 $ $4200$ $12$ $( 1, 6,11,16,26,31)( 2, 9,15,18,27,34, 5, 8,12,19,30,33)( 3, 7,14,20,28,32, 4, 10,13,17,29,35)(22,24,25,23)$
$ 12, 12, 6, 4, 1 $ $4200$ $12$ $( 1, 6,11,16,26,31)( 2, 8,15,19,27,33, 5, 9,12,18,30,34)( 3,10,14,17,28,35, 4, 7,13,20,29,32)(22,23,25,24)$
$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $350$ $6$ $( 1,23,31, 3,21,33)( 2,22,32)( 4,25,34, 5,24,35)( 6, 8)( 9,10)(11,13)(14,15) (16,18)(19,20)(26,28)(29,30)$
$ 15, 5, 5, 5, 5 $ $280$ $15$ $( 1,24,32, 5,23,31, 4,22,35, 3,21,34, 2,25,33)( 6, 9, 7,10, 8)(11,14,12,15,13) (16,19,17,20,18)(26,29,27,30,28)$
$ 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 $ $70$ $3$ $( 1,21,31)( 2,22,32)( 3,23,33)( 4,24,34)( 5,25,35)$
$ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ $2100$ $12$ $( 1,33,22, 5,31,23, 2,35,21, 3,32,25)( 4,34,24)( 6, 8, 7,10)(11,28,12,30) (13,27,15,26)(14,29)(16,18,17,20)$
$ 12, 4, 4, 4, 4, 3, 2, 1, 1 $ $2100$ $12$ $( 1,34,23, 5,31,24, 3,35,21, 4,33,25)( 2,32,22)( 6, 9, 8,10)(11,29,13,30) (12,27)(14,28,15,26)(16,19,18,20)$
$ 10, 10, 5, 5, 5 $ $420$ $10$ $( 1, 5, 4, 3, 2)( 6,20, 9,18, 7,16,10,19, 8,17)(11,30,14,28,12,26,15,29,13,27) (21,25,24,23,22)(31,35,34,33,32)$
$ 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 $ $105$ $2$ $( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,26)(12,27)(13,28)(14,29)(15,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $525$ $2$ $( 1, 2)( 3, 5)( 6,17)( 7,16)( 8,20)( 9,19)(10,18)(11,27)(12,26)(13,30)(14,29) (15,28)(21,22)(23,25)(31,32)(33,35)$
$ 15, 10, 10 $ $840$ $30$ $( 1,24,32, 5,23,31, 4,22,35, 3,21,34, 2,25,33)( 6,19, 7,20, 8,16, 9,17,10,18) (11,29,12,30,13,26,14,27,15,28)$
$ 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $210$ $6$ $( 1,21,31)( 2,22,32)( 3,23,33)( 4,24,34)( 5,25,35)( 6,16)( 7,17)( 8,18)( 9,19) (10,20)(11,26)(12,27)(13,28)(14,29)(15,30)$
$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1050$ $6$ $( 1,23,31, 3,21,33)( 2,22,32)( 4,25,34, 5,24,35)( 6,18)( 7,17)( 8,16)( 9,20) (10,19)(11,28)(12,27)(13,26)(14,30)(15,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $1050$ $4$ $( 1, 2, 5, 4)( 6,17,30,14)( 7,20,29,11)( 8,18,28,13)( 9,16,27,15)(10,19,26,12) (21,22,25,24)(31,32,35,34)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $1050$ $4$ $( 1, 5, 3, 4)( 6,20,28,14)( 7,17,27,12)( 8,19,26,15)( 9,16,30,13)(10,18,29,11) (21,25,23,24)(31,35,33,34)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ $3150$ $4$ $( 1,32)( 2,31)( 3,35)( 4,34)( 5,33)( 6,12,16,27)( 7,11,17,26)( 8,15,18,30) ( 9,14,19,29)(10,13,20,28)(21,22)(23,25)$
$ 20, 10, 5 $ $2520$ $20$ $( 1,35, 4,33, 2,31, 5,34, 3,32)( 6,15,19,28, 7,11,20,29, 8,12,16,30, 9,13,17, 26,10,14,18,27)(21,25,24,23,22)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $630$ $4$ $( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,11,16,26)( 7,12,17,27)( 8,13,18,28) ( 9,14,19,29)(10,15,20,30)$
$ 12, 4, 4, 4, 4, 4, 3 $ $2100$ $12$ $( 1,21,31)( 2,23,35, 4,22,33, 5,24,32, 3,25,34)( 6,16,26,11)( 7,18,30,14) ( 8,20,29,12)( 9,17,28,15)(10,19,27,13)$
$ 12, 4, 4, 4, 4, 4, 3 $ $2100$ $12$ $( 1,21,31)( 2,24,35, 3,22,34, 5,23,32, 4,25,33)( 6,16,26,11)( 7,19,30,13) ( 8,17,29,15)( 9,20,28,12)(10,18,27,14)$

Group invariants

Order:  $50400=2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.