Properties

Label 35T27
Order \(2520\)
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 $t$ :  $27$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,14,5,33,13,2,31,12,4,35,15,3,32,11)(6,16,22,9,20,21,8,18,23,7,19,25,10,17,24)(26,27,28,30,29), (1,12,27,35,22,9)(2,15,30,34,21,6)(3,14,29,31,25,8)(4,13,26,33,24,10)(5,11,28,32,23,7)(17,20,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
42:  $F_7$
60:  $A_5$
120:  $A_5\times C_2$
180:  $\GL(2,4)$
360:  30T87

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $A_5$

Degree 7: $F_7$

Low degree siblings

42T297

Siblings are shown 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 $ $1$ $1$ $()$
$ 7, 7, 7, 7, 7 $ $6$ $7$ $( 1,13,24,35, 7,20,28)( 2,15,21,34, 6,19,30)( 3,14,25,31, 8,16,29) ( 4,11,23,33, 9,17,27)( 5,12,22,32,10,18,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,21,15)( 7,24,13)( 8,25,14)( 9,23,11)(10,22,12)(16,29,31)(17,27,33) (18,26,32)(19,30,34)(20,28,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $7$ $3$ $( 6,15,21)( 7,13,24)( 8,14,25)( 9,11,23)(10,12,22)(16,31,29)(17,33,27) (18,32,26)(19,34,30)(20,35,28)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,30,21,34,15,19)( 7,28,24,35,13,20)( 8,29,25,31,14,16)( 9,27,23,33,11,17) (10,26,22,32,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $7$ $2$ $( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$
$ 6, 6, 6, 6, 6, 1, 1, 1, 1, 1 $ $7$ $6$ $( 6,19,15,34,21,30)( 7,20,13,35,24,28)( 8,16,14,31,25,29)( 9,17,11,33,23,27) (10,18,12,32,22,26)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1,15,35, 2,13,34)( 3,12,31, 5,14,32)( 4,11,33)( 6,24,19, 7,21,20) ( 8,22,16,10,25,18)( 9,23,17)(26,29)(28,30)$
$ 14, 14, 7 $ $90$ $14$ $( 1, 6,13,19,24,30,35, 2, 7,15,20,21,28,34)( 3,10,14,18,25,26,31, 5, 8,12,16, 22,29,32)( 4, 9,11,17,23,27,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 2)( 3, 5)( 6, 7)( 8,10)(12,14)(13,15)(16,18)(19,20)(21,24)(22,25)(26,29) (28,30)(31,32)(34,35)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $105$ $6$ $( 1,21,35, 2,24,34)( 3,22,31, 5,25,32)( 4,23,33)( 6, 7)( 8,10)(11,27,17) (12,29,18,14,26,16)(13,30,20,15,28,19)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1,19,28,21, 7,34)( 2,20,30,24, 6,35)( 3,18,29,22, 8,32)( 4,17,27,23, 9,33) ( 5,16,26,25,10,31)(12,14)(13,15)$
$ 6, 6, 6, 6, 6, 2, 2, 1 $ $105$ $6$ $( 1,30,13, 6,20,34)( 2,28,15, 7,19,35)( 3,26,14,10,16,32)( 4,27,11, 9,17,33) ( 5,29,12, 8,18,31)(21,24)(22,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $105$ $2$ $( 1,34)( 2,35)( 3,32)( 4,33)( 5,31)( 6,28)( 7,30)( 8,26)( 9,27)(10,29)(11,23) (12,25)(13,21)(14,22)(15,24)(16,18)(19,20)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1,15,32)( 2,12,35)( 3,14,31)( 4,11,33)( 5,13,34)( 6,22,20)( 7,21,18) ( 8,25,16)( 9,23,17)(10,24,19)(26,28,30)$
$ 21, 7, 7 $ $120$ $21$ $( 1, 6,12,20,21,26,35, 2,10,13,19,22,28,34, 5, 7,15,18,24,30,32) ( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 2, 5)( 6,10, 7)(12,13,15)(18,20,19)(21,22,24)(26,28,30)(32,35,34)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $140$ $3$ $( 1,21,32)( 2,22,35)( 3,25,31)( 4,23,33)( 5,24,34)( 6,10, 7)(11,27,17) (12,28,19)(13,30,18)(14,29,16)(15,26,20)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1,19,26,24, 6,32)( 2,18,28,21,10,35)( 3,16,29,25, 8,31)( 4,17,27,23, 9,33) ( 5,20,30,22, 7,34)(12,13,15)$
$ 6, 6, 6, 6, 6, 3, 1, 1 $ $140$ $6$ $( 1,30,12, 7,19,32)( 2,26,13, 6,18,35)( 3,29,14, 8,16,31)( 4,27,11, 9,17,33) ( 5,28,15,10,20,34)(21,22,24)$
$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $ $140$ $6$ $( 1,34, 5,35, 2,32)( 3,31)( 4,33)( 6,26, 7,30,10,28)( 8,29)( 9,27)(11,23) (12,24,15,22,13,21)(14,25)(18,20,19)$
$ 30, 5 $ $84$ $30$ $( 1,15,26,31,23, 7, 2,12,29,33,24, 6, 5,14,27,35,21,10, 3,11,28,34,22, 8, 4, 13,30,32,25, 9)(16,17,20,19,18)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1, 6, 5, 8, 4, 7, 2,10, 3, 9)(11,35,15,32,14,33,13,34,12,31)(16,27,20,30,18, 29,17,28,19,26)(21,22,25,23,24)$
$ 30, 5 $ $84$ $30$ $( 1,21,12,16,33, 7, 2,22,14,17,35, 6, 5,25,11,20,34,10, 3,23,13,19,32, 8, 4, 24,15,18,31, 9)(26,29,27,28,30)$
$ 15, 15, 5 $ $84$ $15$ $( 1,19,10, 3,17, 7, 2,18, 8, 4,20, 6, 5,16, 9)(11,24,30,12,25,27,13,21,26,14, 23,28,15,22,29)(31,33,35,34,32)$
$ 15, 15, 5 $ $84$ $15$ $( 1,30,10, 3,27, 7, 2,26, 8, 4,28, 6, 5,29, 9)(11,13,15,12,14)(16,23,35,19,22, 31,17,24,34,18,25,33,20,21,32)$
$ 35 $ $72$ $35$ $( 1,34,26,25,17,13, 6, 5,31,27,24,19,12, 8, 4,35,30,22,16,11, 7, 2,32,29,23, 20,15,10, 3,33,28,21,18,14, 9)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 5, 3, 4)( 6,10, 8, 9, 7)(11,13,15,12,14)(16,17,20,19,18) (21,22,25,23,24)(26,29,27,28,30)(31,33,35,34,32)$
$ 15, 15, 5 $ $84$ $15$ $( 1,15,32, 4,14,35, 2,12,33, 3,13,34, 5,11,31)( 6,22,17, 8,24,19,10,23,16, 7, 21,18, 9,25,20)(26,27,29,28,30)$
$ 35 $ $72$ $35$ $( 1, 6,12,17,25,28,34, 5, 9,14,20,21,26,33, 3, 7,15,18,23,29,35, 2,10,11,16, 24,30,32, 4, 8,13,19,22,27,31)$
$ 5, 5, 5, 5, 5, 5, 5 $ $12$ $5$ $( 1, 2, 5, 4, 3)( 6,10, 9, 8, 7)(11,14,13,15,12)(16,20,19,18,17) (21,22,23,25,24)(26,27,29,28,30)(31,35,34,32,33)$
$ 15, 15, 5 $ $84$ $15$ $( 1,21,32, 4,25,35, 2,22,33, 3,24,34, 5,23,31)( 6,10, 9, 8, 7)(11,29,20,15,26, 17,14,28,19,12,27,16,13,30,18)$
$ 30, 5 $ $84$ $30$ $( 1,19,26,23, 8,35, 2,18,27,25, 7,34, 5,17,29,24, 6,32, 4,16,28,21,10,33, 3, 20,30,22, 9,31)(11,14,13,15,12)$
$ 30, 5 $ $84$ $30$ $( 1,30,12, 9,16,35, 2,26,11, 8,20,34, 5,27,14, 7,19,32, 4,29,13, 6,18,33, 3, 28,15,10,17,31)(21,22,23,25,24)$
$ 10, 10, 10, 5 $ $84$ $10$ $( 1,34, 5,33, 3,35, 2,32, 4,31)( 6,26, 9,29, 7,30,10,27, 8,28)(11,25,13,21,12, 23,14,24,15,22)(16,20,19,18,17)$

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.