Properties

Label 21T41
Order \(6174\)
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 $t$ :  $41$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8,4,11,2,9)(3,10,5,12,6,13)(7,14)(15,18,19,17,21,20), (1,10,18,2,9,16,3,8,21,4,14,19,5,13,17,6,12,15,7,11,20)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
42:  $F_7$
126:  21T10
882:  14T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

21T41 x 5, 42T465 x 6, 42T472 x 3, 42T475 x 2

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 $ $18$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 1, 5, 2, 6, 3, 7, 4)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
$ 7, 7, 7 $ $18$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
$ 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 $ $294$ $21$ $( 1,19, 8, 2,15,11, 3,18,14, 4,21,10, 5,17,13, 6,20, 9, 7,16,12)$
$ 21 $ $294$ $21$ $( 1,21, 8, 4,16,10, 7,18,12, 3,20,14, 6,15, 9, 2,17,11, 5,19,13)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1,18, 8)( 2,21,11)( 3,17,14)( 4,20,10)( 5,16,13)( 6,19, 9)( 7,15,12)$
$ 21 $ $294$ $21$ $( 1,17,14, 6,21, 9, 4,18,11, 2,15,13, 7,19, 8, 5,16,10, 3,20,12)$
$ 21 $ $294$ $21$ $( 1,19,13, 2,17,12, 3,15,11, 4,20,10, 5,18, 9, 6,16, 8, 7,21,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1,16,11)( 2,21,10)( 3,19, 9)( 4,17, 8)( 5,15,14)( 6,20,13)( 7,18,12)$
$ 21 $ $294$ $21$ $( 1,16, 9, 6,18,13, 4,20,10, 2,15,14, 7,17,11, 5,19, 8, 3,21,12)$
$ 21 $ $294$ $21$ $( 1,17,10, 3,15,13, 5,20, 9, 7,18,12, 2,16, 8, 4,21,11, 6,19,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $98$ $3$ $( 1,20,13)( 2,19,11)( 3,18, 9)( 4,17,14)( 5,16,12)( 6,15,10)( 7,21, 8)$
$ 6, 6, 6, 2, 1 $ $1029$ $6$ $( 2, 6, 5, 7, 3, 4)( 8,20,12,19,14,15)( 9,18)(10,16,13,17,11,21)$
$ 14, 2, 2, 2, 1 $ $441$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,21, 9,20,10,19,11,18,12,17,13,16,14,15)$
$ 14, 2, 2, 2, 1 $ $441$ $14$ $( 1, 2)( 3, 7)( 4, 6)( 8,15,14,16,13,17,12,18,11,19,10,20, 9,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $147$ $2$ $( 1, 5)( 2, 4)( 6, 7)( 8,18)( 9,17)(10,16)(11,15)(12,21)(13,20)(14,19)$
$ 6, 6, 6, 2, 1 $ $1029$ $6$ $( 2, 4, 3, 7, 5, 6)( 8,18,10,17,14,15)( 9,21,12,16,11,20)(13,19)$

Group invariants

Order:  $6174=2 \cdot 3^{2} \cdot 7^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.