Properties

Label 21T6
Degree $21$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times C_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 6);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times C_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  42.3
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);