Properties

Label 35T35
Order \(20160\)
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 $t$ :  $35$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,21,11,30,10,19,35,5,25,15,26,8,18,34,2,22,14,28,9,16,33,3,24,12,27,6,20,32)(4,23,13,29,7,17,31), (1,29,35)(2,27,31)(3,30,34,5,28,32)(4,26,33)(6,12,21,8,15,22)(7,11,25)(9,14,23)(10,13,24)(16,19)(17,20,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
120:  $S_5$
168:  $\GL(3,2)$
336:  14T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $S_5$

Degree 7: $\GL(3,2)$

Low degree siblings

35T35, 40T11250, 42T723 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 1, 5)( 8,10)(12,14)(16,18)(22,25)(27,30)(32,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $15$ $2$ $( 1, 5)( 2, 4)( 7, 9)( 8,10)(11,13)(12,14)(16,18)(17,20)(22,25)(23,24)(26,29) (27,30)(31,35)(32,33)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $3$ $( 1, 5, 4)( 7,10, 8)(12,13,14)(16,17,18)(22,23,25)(27,30,29)(31,33,32)$
$ 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $ $20$ $6$ $( 1, 5, 4)( 2, 3)( 6, 9)( 7,10, 8)(11,15)(12,13,14)(16,17,18)(19,20)(21,24) (22,23,25)(26,28)(27,30,29)(31,33,32)(34,35)$
$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ $30$ $4$ $( 1, 5, 4, 2)( 7, 9,10, 8)(11,14,12,13)(16,17,20,18)(22,23,24,25)(26,27,30,29) (31,35,33,32)$
$ 5, 5, 5, 5, 5, 5, 5 $ $24$ $5$ $( 1, 5, 4, 2, 3)( 6,10, 8, 7, 9)(11,15,14,12,13)(16,17,20,19,18) (21,25,22,23,24)(26,28,27,30,29)(31,35,34,33,32)$
$ 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 $ $21$ $2$ $(11,24)(12,22)(13,23)(14,25)(15,21)(26,35)(27,33)(28,34)(29,31)(30,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $210$ $2$ $( 1, 5)( 8,10)(11,24)(12,25)(13,23)(14,22)(15,21)(16,18)(26,35)(27,32)(28,34) (29,31)(30,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $315$ $2$ $( 1, 5)( 2, 4)( 7, 9)( 8,10)(11,23)(12,25)(13,24)(14,22)(15,21)(16,18)(17,20) (26,31)(27,32)(28,34)(29,35)(30,33)$
$ 6, 6, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $420$ $6$ $( 1, 5, 4)( 7,10, 8)(11,24)(12,23,14,22,13,25)(15,21)(16,17,18)(26,35) (27,32,29,33,30,31)(28,34)$
$ 6, 6, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $ $420$ $6$ $( 1, 5, 4)( 2, 3)( 6, 9)( 7,10, 8)(11,21)(12,23,14,22,13,25)(15,24)(16,17,18) (19,20)(26,34)(27,32,29,33,30,31)(28,35)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 1, 1, 1 $ $630$ $4$ $( 1, 5, 4, 2)( 7, 9,10, 8)(11,25,12,23)(13,24,14,22)(15,21)(16,17,20,18) (26,33,30,31)(27,32,29,35)(28,34)$
$ 10, 10, 5, 5, 5 $ $504$ $10$ $( 1, 5, 4, 2, 3)( 6,10, 8, 7, 9)(11,21,14,22,13,24,15,25,12,23) (16,17,20,19,18)(26,34,27,32,29,35,28,33,30,31)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $42$ $4$ $( 6,15,19,34)( 7,13,17,31)( 8,12,16,32)( 9,11,20,35)(10,14,18,33)(21,28) (22,30)(23,29)(24,26)(25,27)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $420$ $4$ $( 1, 5)( 6,15,19,34)( 7,13,17,31)( 8,14,16,33)( 9,11,20,35)(10,12,18,32) (21,28)(22,27)(23,29)(24,26)(25,30)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ $630$ $4$ $( 1, 5)( 2, 4)( 6,15,19,34)( 7,11,17,35)( 8,14,16,33)( 9,13,20,31) (10,12,18,32)(21,28)(22,27)(23,26)(24,29)(25,30)$
$ 12, 6, 4, 4, 3, 2, 2, 1, 1 $ $840$ $12$ $( 1, 5, 4)( 6,15,19,34)( 7,14,16,31,10,12,17,33, 8,13,18,32)( 9,11,20,35) (21,28)(22,29,25,30,23,27)(24,26)$
$ 12, 6, 4, 4, 3, 2, 2, 2 $ $840$ $12$ $( 1, 5, 4)( 2, 3)( 6,11,19,35)( 7,14,16,31,10,12,17,33, 8,13,18,32) ( 9,15,20,34)(21,26)(22,29,25,30,23,27)(24,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $1260$ $4$ $( 1, 5, 4, 2)( 6,15,19,34)( 7,11,18,32)( 8,13,20,33)( 9,14,16,31)(10,12,17,35) (21,28)(22,29,24,27)(23,26,25,30)$
$ 20, 10, 5 $ $1008$ $20$ $( 1, 5, 4, 2, 3)( 6,14,16,31, 9,15,18,32, 7,11,19,33, 8,13,20,34,10,12,17,35) (21,27,22,29,24,28,25,30,23,26)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $56$ $3$ $( 6,15,21)( 7,13,23)( 8,12,22)( 9,11,24)(10,14,25)(16,32,30)(17,31,29) (18,33,27)(19,34,28)(20,35,26)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 2, 1, 1, 1 $ $560$ $6$ $( 1, 5)( 6,15,21)( 7,13,23)( 8,14,22,10,12,25)( 9,11,24)(16,33,30,18,32,27) (17,31,29)(19,34,28)(20,35,26)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $840$ $6$ $( 1, 5)( 2, 4)( 6,15,21)( 7,11,23, 9,13,24)( 8,14,22,10,12,25)(16,33,30,18,32, 27)(17,35,29,20,31,26)(19,34,28)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $1120$ $3$ $( 1, 5, 4)( 6,15,21)( 7,14,22)( 8,13,25)( 9,11,24)(10,12,23)(16,31,27) (17,33,30)(18,32,29)(19,34,28)(20,35,26)$
$ 6, 6, 3, 3, 3, 3, 3, 3, 3, 2 $ $1120$ $6$ $( 1, 5, 4)( 2, 3)( 6,11,21, 9,15,24)( 7,14,22)( 8,13,25)(10,12,23)(16,31,27) (17,33,30)(18,32,29)(19,35,28,20,34,26)$
$ 12, 12, 4, 3, 3, 1 $ $1680$ $12$ $( 1, 5, 4, 2)( 6,15,21)( 7,11,25, 8,13,24,10,12,23, 9,14,22)(16,31,26,18,32, 29,20,33,30,17,35,27)(19,34,28)$
$ 15, 15, 5 $ $1344$ $15$ $( 1, 5, 4, 2, 3)( 6,14,22, 7,11,21,10,12,23, 9,15,25, 8,13,24)(16,31,26,19,33, 30,17,35,28,18,32,29,20,34,27)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,10,14,18,25,27,33)( 2, 9,11,20,24,26,35)( 3, 6,15,19,21,28,34) ( 4, 7,13,17,23,29,31)( 5, 8,12,16,22,30,32)$
$ 14, 7, 7, 7 $ $240$ $14$ $( 1, 8,14,16,25,30,33, 5,10,12,18,22,27,32)( 2, 9,11,20,24,26,35) ( 3, 6,15,19,21,28,34)( 4, 7,13,17,23,29,31)$
$ 14, 14, 7 $ $360$ $14$ $( 1, 8,14,16,25,30,33, 5,10,12,18,22,27,32)( 2, 7,11,17,24,29,35, 4, 9,13,20, 23,26,31)( 3, 6,15,19,21,28,34)$
$ 21, 7, 7 $ $480$ $21$ $( 1, 8,13,18,22,29,33, 5, 7,14,16,23,27,32, 4,10,12,17,25,30,31) ( 2, 9,11,20,24,26,35)( 3, 6,15,19,21,28,34)$
$ 21, 14 $ $480$ $42$ $( 1, 8,13,18,22,29,33, 5, 7,14,16,23,27,32, 4,10,12,17,25,30,31) ( 2, 6,11,19,24,28,35, 3, 9,15,20,21,26,34)$
$ 28, 7 $ $720$ $28$ $( 1, 8,13,20,25,30,31, 2,10,12,17,24,27,32, 4, 9,14,16,23,26,33, 5, 7,11,18, 22,29,35)( 3, 6,15,19,21,28,34)$
$ 35 $ $576$ $35$ $( 1, 8,13,20,21,27,32, 4, 9,15,18,22,29,35, 3,10,12,17,24,28,33, 5, 7,11,19, 25,30,31, 2, 6,14,16,23,26,34)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1,10,14,33,27,18,25)( 2, 9,11,35,26,20,24)( 3, 6,15,34,28,19,21) ( 4, 7,13,31,29,17,23)( 5, 8,12,32,30,16,22)$
$ 14, 7, 7, 7 $ $240$ $14$ $( 1, 8,14,32,27,16,25, 5,10,12,33,30,18,22)( 2, 9,11,35,26,20,24) ( 3, 6,15,34,28,19,21)( 4, 7,13,31,29,17,23)$
$ 14, 14, 7 $ $360$ $14$ $( 1, 8,14,32,27,16,25, 5,10,12,33,30,18,22)( 2, 7,11,31,26,17,24, 4, 9,13,35, 29,20,23)( 3, 6,15,34,28,19,21)$
$ 21, 7, 7 $ $480$ $21$ $( 1, 8,13,33,30,17,25, 5, 7,14,32,29,18,22, 4,10,12,31,27,16,23) ( 2, 9,11,35,26,20,24)( 3, 6,15,34,28,19,21)$
$ 21, 14 $ $480$ $42$ $( 1, 8,13,33,30,17,25, 5, 7,14,32,29,18,22, 4,10,12,31,27,16,23) ( 2, 6,11,34,26,19,24, 3, 9,15,35,28,20,21)$
$ 28, 7 $ $720$ $28$ $( 1, 8,13,35,27,16,23, 2,10,12,31,26,18,22, 4, 9,14,32,29,20,25, 5, 7,11,33, 30,17,24)( 3, 6,15,34,28,19,21)$
$ 35 $ $576$ $35$ $( 1, 8,13,35,28,18,22, 4, 9,15,33,30,17,24, 3,10,12,31,26,19,25, 5, 7,11,34, 27,16,23, 2, 6,14,32,29,20,21)$

Group invariants

Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.