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

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.