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

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.