Properties

Label 21T35
Order \(3087\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $35$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,16,5,10,21,7,9,20)(2,8,19)(3,11,15,6,13,17,4,14,18), (1,5,2,6,3,7,4)(8,11,14,10,13,9,12)(15,16,17,18,19,20,21)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T35 x 18

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 $ $1$ $1$ $()$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 3, 5, 7, 2, 4, 6)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $343$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $343$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 9, 9, 3 $ $343$ $9$ $( 1,12,16, 5,10,21, 7, 9,20)( 2, 8,19)( 3,11,15, 6,13,17, 4,14,18)$
$ 9, 9, 3 $ $343$ $9$ $( 1,10,18, 2, 8,17, 6,14,20)( 3,13,16)( 4,11,15, 7,12,19, 5, 9,21)$
$ 9, 9, 3 $ $343$ $9$ $( 1, 9,18, 5,12,17, 7,10,20)( 2, 8,16)( 3,14,21, 6,11,15, 4,13,19)$
$ 9, 9, 3 $ $343$ $9$ $( 1,16,10, 7,20,12, 5,21, 9)( 2,19, 8)( 3,15,13, 4,18,11, 6,17,14)$
$ 9, 9, 3 $ $343$ $9$ $( 1,19, 8, 5,18,13, 6,16, 9)( 2,17,11, 7,21,12, 3,15,14)( 4,20,10)$
$ 9, 9, 3 $ $343$ $9$ $( 1,17,13, 7,18,14, 5,20, 9)( 2,16,12)( 3,15,11, 4,21,10, 6,19, 8)$

Group invariants

Order:  $3087=3^{2} \cdot 7^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.