Properties

Label 21T49
Order \(9261\)
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 $t$ :  $49$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,17)(2,12,15,3,13,20,5,8,16)(4,14,18,7,10,19,6,9,21), (1,7,3)(2,4,5)(8,11,14,10,13,9,12)(15,16,18)(17,20,19)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
27:  $C_9:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

There are no siblings 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 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 8,11,14,10,13, 9,12)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $343$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $343$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,11,17)( 2,12,15, 3,13,20, 5, 8,16)( 4,14,18, 7,10,19, 6, 9,21)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,17,11)( 2,15,13, 5,16,12, 3,20, 8)( 4,18,10, 6,21,14, 7,19, 9)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $147$ $3$ $( 9,10,12)(11,14,13)(16,19,17)(18,20,21)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12,13)( 9,14,10)(15,19,21)(17,20,18)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13, 9)(11,12,14)(15,20,19)(16,17,21)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,11,18, 5, 8,16, 6, 9,19)( 2,12,21, 7,10,15, 3,13,17)( 4,14,20)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,17,12, 4,18, 8, 2,15,13)( 3,20,14, 5,16, 9, 6,21,10)( 7,19,11)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $147$ $3$ $( 9,12,10)(11,13,14)(16,17,19)(18,21,20)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12,14)(10,13,11)(15,19,20)(16,21,17)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,12)( 9,10,14)(15,20,16)(18,19,21)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,11,20, 7,10,21, 5, 8,16)( 2,12,19)( 3,13,18, 4,14,17, 6, 9,15)$
$ 9, 9, 3 $ $1029$ $9$ $( 1,17, 9, 3,20,10, 4,18,14)( 2,15,13, 7,19,12, 6,21, 8)( 5,16,11)$

Group invariants

Order:  $9261=3^{3} \cdot 7^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.