Properties

Label 21T48
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 $t$ :  $48$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,5)(2,3,7)(8,12,13)(9,14,10), (1,15,9,3,20,10,5,18,11,7,16,12,2,21,13,4,19,14,6,17,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T48 x 3

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 6, 4, 2, 7, 5, 3)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 6, 4, 2, 7, 5, 3)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $27$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$ 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, 7 $ $9$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $147$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)(15,17,19,21,16,18,20)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)(15,21,20,19,18,17,16)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $147$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)(15,17,19,21,16,18,20)$
$ 7, 3, 3, 3, 3, 1, 1 $ $441$ $21$ $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)(15,21,20,19,18,17,16)$
$ 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 $ $343$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 21 $ $441$ $21$ $( 1,15, 9, 3,20,10, 5,18,11, 7,16,12, 2,21,13, 4,19,14, 6,17, 8)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1,15, 9)( 2,21,13)( 3,20,10)( 4,19,14)( 5,18,11)( 6,17, 8)( 7,16,12)$
$ 21 $ $441$ $21$ $( 1,15, 9, 4,19,14, 7,16,12, 3,20,10, 6,17, 8, 2,21,13, 5,18,11)$
$ 21 $ $441$ $21$ $( 1,15,10, 3,20,12, 5,18,14, 7,16, 9, 2,21,11, 4,19,13, 6,17, 8)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1,15,10)( 2,21,11)( 3,20,12)( 4,19,13)( 5,18,14)( 6,17, 8)( 7,16, 9)$
$ 21 $ $441$ $21$ $( 1,15,10, 4,19,13, 7,16, 9, 3,20,12, 6,17, 8, 2,21,11, 5,18,14)$
$ 21 $ $441$ $21$ $( 1,15,12, 3,20, 9, 5,18,13, 7,16,10, 2,21,14, 4,19,11, 6,17, 8)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1,15,12)( 2,21,14)( 3,20, 9)( 4,19,11)( 5,18,13)( 6,17, 8)( 7,16,10)$
$ 21 $ $441$ $21$ $( 1,15,12, 4,19,11, 7,16,10, 3,20, 9, 6,17, 8, 2,21,14, 5,18,13)$
$ 21 $ $441$ $21$ $( 1, 9,20, 5,11,16, 2,13,19, 6, 8,15, 3,10,18, 7,12,21, 4,14,17)$
$ 21 $ $441$ $21$ $( 1, 9,20, 6, 8,15, 4,14,17, 2,13,19, 7,12,21, 5,11,16, 3,10,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1, 9,20)( 2,13,19)( 3,10,18)( 4,14,17)( 5,11,16)( 6, 8,15)( 7,12,21)$
$ 21 $ $441$ $21$ $( 1,10,18, 4,13,19, 7, 9,20, 3,12,21, 6, 8,15, 2,11,16, 5,14,17)$
$ 21 $ $441$ $21$ $( 1,10,18, 2,11,16, 3,12,21, 4,13,19, 5,14,17, 6, 8,15, 7, 9,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1,10,18)( 2,11,16)( 3,12,21)( 4,13,19)( 5,14,17)( 6, 8,15)( 7, 9,20)$
$ 21 $ $441$ $21$ $( 1,12,21, 7,10,18, 6, 8,15, 5,13,19, 4,11,16, 3, 9,20, 2,14,17)$
$ 21 $ $441$ $21$ $( 1,12,21, 5,13,19, 2,14,17, 6, 8,15, 3, 9,20, 7,10,18, 4,11,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $147$ $3$ $( 1,12,21)( 2,14,17)( 3, 9,20)( 4,11,16)( 5,13,19)( 6, 8,15)( 7,10,18)$

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.