Properties

Label 21T6
Order \(42\)
n \(21\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7\times S_3$

Related objects

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Group action invariants

Degree $n$ :  $21$
Transitive number $t$ :  $6$
Group :  $C_7\times S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21), (1,12,19,9,16,6,13,3,10,21,7,18,4,15)(2,11,20,8,17,5,14)
$|\Aut(F/K)|$:  $7$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
7:  $C_7$
14:  $C_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $C_7$

Low degree siblings

42T6

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

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [42, 3]
Character table: Data not available.