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Properties

Label 21T56
Degree $21$
Order $15120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_3\times S_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 56);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $56$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times S_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,17)(2,5,18)(3,6,16)(7,14,19,12,9,13,21,11,8,15,20,10), (4,11,14,18,20,9)(5,12,15,16,21,7)(6,10,13,17,19,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$5040$:  $S_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $S_7$

Low degree siblings

42T666

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ $504$ $5$ $( 1, 7,21,12,15)( 2, 8,19,10,13)( 3, 9,20,11,14)$
$ 15, 3, 3 $ $504$ $15$ $( 1, 8,20,12,13, 3, 7,19,11,15, 2, 9,21,10,14)( 4, 5, 6)(16,17,18)$
$ 15, 3, 3 $ $504$ $15$ $( 1, 9,19,12,14, 2, 7,20,10,15, 3, 8,21,11,13)( 4, 6, 5)(16,18,17)$
$ 6, 3, 3, 3, 3, 3 $ $21$ $6$ $( 1, 2, 3)( 4,16, 6,18, 5,17)( 7, 8, 9)(10,11,12)(13,14,15)(19,20,21)$
$ 6, 3, 3, 3, 3, 3 $ $21$ $6$ $( 1, 3, 2)( 4,17, 5,18, 6,16)( 7, 9, 8)(10,12,11)(13,15,14)(19,21,20)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 4,18)( 5,16)( 6,17)$
$ 5, 5, 5, 2, 2, 2 $ $504$ $10$ $( 1,21,15, 7,12)( 2,19,13, 8,10)( 3,20,14, 9,11)( 4,18)( 5,16)( 6,17)$
$ 15, 6 $ $504$ $30$ $( 1,19,14, 7,10, 3,21,13, 9,12, 2,20,15, 8,11)( 4,16, 6,18, 5,17)$
$ 15, 6 $ $504$ $30$ $( 1,20,13, 7,11, 2,21,14, 8,12, 3,19,15, 9,10)( 4,17, 5,18, 6,16)$
$ 21 $ $720$ $21$ $( 1, 6,11,21, 8,18,15, 2, 4,12,19, 9,16,13, 3, 5,10,20, 7,17,14)$
$ 21 $ $720$ $21$ $( 1, 4,10,21, 9,17,15, 3, 6,12,20, 8,16,14, 2, 5,11,19, 7,18,13)$
$ 7, 7, 7 $ $720$ $7$ $( 1, 5,12,21, 7,16,15)( 2, 6,10,19, 8,17,13)( 3, 4,11,20, 9,18,14)$
$ 6, 6, 3, 3, 3 $ $105$ $6$ $( 1, 3, 2)( 4, 6, 5)( 7,11, 8,12, 9,10)(13,15,14)(16,20,17,21,18,19)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 7,12)( 8,10)( 9,11)(16,21)(17,19)(18,20)$
$ 6, 6, 3, 3, 3 $ $105$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,10, 9,12, 8,11)(13,14,15)(16,19,18,21,17,20)$
$ 12, 6, 3 $ $630$ $12$ $( 1, 4, 2, 5, 3, 6)( 7,18,10,21, 9,17,12,20, 8,16,11,19)(13,15,14)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $630$ $4$ $( 1, 5)( 2, 6)( 3, 4)( 7,16,12,21)( 8,17,10,19)( 9,18,11,20)$
$ 12, 6, 3 $ $630$ $12$ $( 1, 6, 3, 5, 2, 4)( 7,17,11,21, 8,18,12,19, 9,16,10,20)(13,14,15)$
$ 12, 3, 3, 3 $ $210$ $12$ $( 1, 3, 2)( 4, 6, 5)( 7,20,10,16, 9,19,12,18, 8,21,11,17)(13,15,14)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $210$ $4$ $( 7,21,12,16)( 8,19,10,17)( 9,20,11,18)$
$ 12, 3, 3, 3 $ $210$ $12$ $( 1, 2, 3)( 4, 5, 6)( 7,19,11,16, 8,20,12,17, 9,21,10,18)(13,14,15)$
$ 6, 6, 6, 3 $ $105$ $6$ $( 1, 3, 2)( 4,13, 5,14, 6,15)( 7,11, 8,12, 9,10)(16,20,17,21,18,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 4,14)( 5,15)( 6,13)( 7,12)( 8,10)( 9,11)(16,21)(17,19)(18,20)$
$ 6, 6, 6, 3 $ $105$ $6$ $( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,10, 9,12, 8,11)(16,19,18,21,17,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $280$ $3$ $( 4, 9,20)( 5, 7,21)( 6, 8,19)(10,17,13)(11,18,14)(12,16,15)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1, 2, 3)( 4, 7,19)( 5, 8,20)( 6, 9,21)(10,18,15)(11,16,13)(12,17,14)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1, 3, 2)( 4, 8,21)( 5, 9,19)( 6, 7,20)(10,16,14)(11,17,15)(12,18,13)$
$ 6, 6, 6, 3 $ $840$ $6$ $( 1, 3, 2)( 4,17, 7,14,19,12)( 5,18, 8,15,20,10)( 6,16, 9,13,21,11)$
$ 6, 6, 6, 1, 1, 1 $ $840$ $6$ $( 4,18, 9,14,20,11)( 5,16, 7,15,21,12)( 6,17, 8,13,19,10)$
$ 6, 6, 6, 3 $ $840$ $6$ $( 1, 2, 3)( 4,16, 8,14,21,10)( 5,17, 9,15,19,11)( 6,18, 7,13,20,12)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 7,15,21)( 8,13,19)( 9,14,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7,13,20)( 8,14,21)( 9,15,19)(10,11,12)(16,17,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7,14,19)( 8,15,20)( 9,13,21)(10,12,11)(16,18,17)$
$ 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $420$ $6$ $( 1,12)( 2,10)( 3,11)( 7,21,15)( 8,19,13)( 9,20,14)$
$ 6, 3, 3, 3, 3, 3 $ $420$ $6$ $( 1,10, 3,12, 2,11)( 4, 5, 6)( 7,19,14)( 8,20,15)( 9,21,13)(16,17,18)$
$ 6, 3, 3, 3, 3, 3 $ $420$ $6$ $( 1,11, 2,12, 3,10)( 4, 6, 5)( 7,20,13)( 8,21,14)( 9,19,15)(16,18,17)$
$ 6, 6, 3, 3, 3 $ $210$ $6$ $( 1,18, 2,16, 3,17)( 4,10, 5,11, 6,12)( 7,20,13)( 8,21,14)( 9,19,15)$
$ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ $210$ $6$ $( 1,16)( 2,17)( 3,18)( 4,11)( 5,12)( 6,10)( 7,21,15)( 8,19,13)( 9,20,14)$
$ 6, 6, 3, 3, 3 $ $210$ $6$ $( 1,17, 3,16, 2,18)( 4,12, 6,11, 5,10)( 7,19,14)( 8,20,15)( 9,21,13)$
$ 4, 4, 4, 3, 3, 3 $ $420$ $12$ $( 1,12,16, 5)( 2,10,17, 6)( 3,11,18, 4)( 7,15,21)( 8,13,19)( 9,14,20)$
$ 12, 3, 3, 3 $ $420$ $12$ $( 1,10,18, 5, 2,11,16, 6, 3,12,17, 4)( 7,13,20)( 8,14,21)( 9,15,19)$
$ 12, 3, 3, 3 $ $420$ $12$ $( 1,11,17, 5, 3,10,16, 4, 2,12,18, 6)( 7,14,19)( 8,15,20)( 9,13,21)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $15120=2^{4} \cdot 3^{3} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  15120.d
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);