Properties

Label 35T21
Order \(840\)
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 $t$ :  $21$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,24,32,10,3,21,34,7,5,23,31,9,2,25,33,6,4,22,35,8)(11,14,12,15,13)(16,29,17,30,18,26,19,27,20,28), (1,17,28,4,20,26,2,18,29,5,16,27,3,19,30)(6,12,23,9,15,21,7,13,24,10,11,22,8,14,25)(31,32,33,34,35)
$|\Aut(F/K)|$:  $5$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $C_5$

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

Low degree siblings

35T21, 40T885

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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,13,15,12,14)(16,18,20,17,19) (21,23,25,22,24)(26,28,30,27,29)(31,33,35,32,34)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,15,14,13,12)(16,20,19,18,17) (21,25,24,23,22)(26,30,29,28,27)(31,35,34,33,32)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
$ 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $(11,21)(12,22)(13,23)(14,24)(15,25)(26,31)(27,32)(28,33)(29,34)(30,35)$
$ 10, 10, 5, 5, 5 $ $21$ $10$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)(11,23,15,22,14,21,13,25,12,24) (16,18,20,17,19)(26,33,30,32,29,31,28,35,27,34)$
$ 10, 10, 5, 5, 5 $ $21$ $10$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)(11,25,14,23,12,21,15,24,13,22) (16,20,19,18,17)(26,35,29,33,27,31,30,34,28,32)$
$ 10, 10, 5, 5, 5 $ $21$ $10$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,24,12,25,13,21,14,22,15,23) (16,19,17,20,18)(26,34,27,35,28,31,29,32,30,33)$
$ 10, 10, 5, 5, 5 $ $21$ $10$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,22,13,24,15,21,12,23,14,25) (16,17,18,19,20)(26,32,28,34,30,31,27,33,29,35)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ $42$ $4$ $( 6,11,16,31)( 7,12,17,32)( 8,13,18,33)( 9,14,19,34)(10,15,20,35)(21,26) (22,27)(23,28)(24,29)(25,30)$
$ 20, 10, 5 $ $42$ $20$ $( 1, 3, 5, 2, 4)( 6,13,20,32, 9,11,18,35, 7,14,16,33,10,12,19,31, 8,15,17,34) (21,28,25,27,24,26,23,30,22,29)$
$ 20, 10, 5 $ $42$ $20$ $( 1, 5, 4, 3, 2)( 6,15,19,33, 7,11,20,34, 8,12,16,35, 9,13,17,31,10,14,18,32) (21,30,24,28,22,26,25,29,23,27)$
$ 20, 10, 5 $ $42$ $20$ $( 1, 4, 2, 5, 3)( 6,14,17,35, 8,11,19,32,10,13,16,34, 7,15,18,31, 9,12,20,33) (21,29,22,30,23,26,24,27,25,28)$
$ 20, 10, 5 $ $42$ $20$ $( 1, 2, 3, 4, 5)( 6,12,18,34,10,11,17,33, 9,15,16,32, 8,14,20,31, 7,13,19,35) (21,27,23,29,25,26,22,28,24,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $56$ $3$ $( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$
$ 15, 15, 5 $ $56$ $15$ $( 1, 3, 5, 2, 4)( 6,13,25, 7,14,21, 8,15,22, 9,11,23,10,12,24)(16,33,30,17,34, 26,18,35,27,19,31,28,20,32,29)$
$ 15, 15, 5 $ $56$ $15$ $( 1, 5, 4, 3, 2)( 6,15,24, 8,12,21,10,14,23, 7,11,25, 9,13,22)(16,35,29,18,32, 26,20,34,28,17,31,30,19,33,27)$
$ 15, 15, 5 $ $56$ $15$ $( 1, 4, 2, 5, 3)( 6,14,22,10,13,21, 9,12,25, 8,11,24, 7,15,23)(16,34,27,20,33, 26,19,32,30,18,31,29,17,35,28)$
$ 15, 15, 5 $ $56$ $15$ $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
$ 35 $ $24$ $35$ $( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$
$ 35 $ $24$ $35$ $( 1,10,14,18,22,26,35, 4, 8,12,16,25,29,33, 2, 6,15,19,23,27,31, 5, 9,13,17, 21,30,34, 3, 7,11,20,24,28,32)$
$ 35 $ $24$ $35$ $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$
$ 35 $ $24$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
$ 7, 7, 7, 7, 7 $ $24$ $7$ $( 1, 6,11,31,26,16,21)( 2, 7,12,32,27,17,22)( 3, 8,13,33,28,18,23) ( 4, 9,14,34,29,19,24)( 5,10,15,35,30,20,25)$
$ 35 $ $24$ $35$ $( 1, 8,15,32,29,16,23, 5, 7,14,31,28,20,22, 4, 6,13,35,27,19,21, 3,10,12,34, 26,18,25, 2, 9,11,33,30,17,24)$
$ 35 $ $24$ $35$ $( 1,10,14,33,27,16,25, 4, 8,12,31,30,19,23, 2, 6,15,34,28,17,21, 5, 9,13,32, 26,20,24, 3, 7,11,35,29,18,22)$
$ 35 $ $24$ $35$ $( 1, 9,12,35,28,16,24, 2,10,13,31,29,17,25, 3, 6,14,32,30,18,21, 4, 7,15,33, 26,19,22, 5, 8,11,34,27,20,23)$
$ 35 $ $24$ $35$ $( 1, 7,13,34,30,16,22, 3, 9,15,31,27,18,24, 5, 6,12,33,29,20,21, 2, 8,14,35, 26,17,23, 4,10,11,32,28,19,25)$

Group invariants

Order:  $840=2^{3} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [840, 134]
Character table: Data not available.