Properties

Label 35T24
Order \(1680\)
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 $t$ :  $24$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,7,25,3,31,9,22,5,33,6,24,2,35,8,21,4,32,10,23)(11,29,12,30,13,26,14,27,15,28)(16,19,17,20,18), (1,26,21)(2,30,22,5,27,25)(3,29,23,4,28,24)(6,16,11)(7,20,12,10,17,15)(8,19,13,9,18,14)(32,35)(33,34)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
10:  $D_{5}$
168:  $\GL(3,2)$
336:  14T17

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: $D_{5}$

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

Low degree siblings

35T24, 40T1570

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 $ $2$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,14,12,15,13)(16,19,17,20,18) (21,24,22,25,23)(26,29,27,30,28)(31,34,32,35,33)$
$ 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
$ 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, 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 $ $42$ $10$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)(11,24,12,25,13,21,14,22,15,23) (16,19,17,20,18)(26,34,27,35,28,31,29,32,30,33)$
$ 10, 10, 5, 5, 5 $ $42$ $10$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,22,13,24,15,21,12,23,14,25) (16,17,18,19,20)(26,32,28,34,30,31,27,33,29,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 2, 5)( 3, 4)( 7,10)( 8, 9)(11,21)(12,25)(13,24)(14,23)(15,22)(17,20)(18,19) (26,31)(27,35)(28,34)(29,33)(30,32)$
$ 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 $ $84$ $20$ $( 1, 4, 2, 5, 3)( 6,14,17,35, 8,11,19,32,10,13,16,34, 7,15,18,31, 9,12,20,33) (21,29,22,30,23,26,24,27,25,28)$
$ 20, 10, 5 $ $84$ $20$ $( 1, 2, 3, 4, 5)( 6,12,18,34,10,11,17,33, 9,15,16,32, 8,14,20,31, 7,13,19,35) (21,27,23,29,25,26,22,28,24,30)$
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 1 $ $210$ $4$ $( 2, 5)( 3, 4)( 6,11,16,31)( 7,15,17,35)( 8,14,18,34)( 9,13,19,33) (10,12,20,32)(21,26)(22,30)(23,29)(24,28)(25,27)$
$ 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 $ $112$ $15$ $( 1, 4, 2, 5, 3)( 6,14,22,10,13,21, 9,12,25, 8,11,24, 7,15,23)(16,34,27,20,33, 26,19,32,30,18,31,29,17,35,28)$
$ 15, 15, 5 $ $112$ $15$ $( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $280$ $6$ $( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$
$ 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 $ $48$ $35$ $( 1, 9,12,20,23,26,34, 2,10,13,16,24,27,35, 3, 6,14,17,25,28,31, 4, 7,15,18, 21,29,32, 5, 8,11,19,22,30,33)$
$ 35 $ $48$ $35$ $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
$ 14, 14, 7 $ $120$ $14$ $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
$ 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 $ $48$ $35$ $( 1, 9,12,35,28,16,24, 2,10,13,31,29,17,25, 3, 6,14,32,30,18,21, 4, 7,15,33, 26,19,22, 5, 8,11,34,27,20,23)$
$ 35 $ $48$ $35$ $( 1, 7,13,34,30,16,22, 3, 9,15,31,27,18,24, 5, 6,12,33,29,20,21, 2, 8,14,35, 26,17,23, 4,10,11,32,28,19,25)$
$ 14, 14, 7 $ $120$ $14$ $( 1, 6,11,31,26,16,21)( 2,10,12,35,27,20,22, 5, 7,15,32,30,17,25) ( 3, 9,13,34,28,19,23, 4, 8,14,33,29,18,24)$

Group invariants

Order:  $1680=2^{4} \cdot 3 \cdot 5 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1680, 926]
Character table: Data not available.