Properties

Label 21T42
Order \(6174\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $42$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,5,7,6,3)(8,9,12,14,13,10)(15,20,21,17,19,18), (1,14,18,6,9,17,7,8,21,3,12,19,5,10,20,4,11,16)(2,13,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T42 x 18, 42T466 x 19

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

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.