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Properties

Label 21T24
Degree $21$
Order $882$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:(C_3\times F_7)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 24);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $24$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:(C_3\times F_7)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5,6)(2,7,3)(9,10,12)(11,14,13)(15,18,17)(16,20,21), (1,8,18,2,9,19)(3,10,20,7,14,17)(4,11,21,6,13,16)(5,12,15)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$18$:  $C_6 \times C_3$
$42$:  $F_7$ x 2
$126$:  21T9 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T24, 42T142 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 $ $18$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $18$ $7$ $( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 6, 4, 2, 7, 5, 3)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1,18, 9)( 2,19, 8)( 3,20,14)( 4,21,13)( 5,15,12)( 6,16,11)( 7,17,10)$
$ 21 $ $42$ $21$ $( 1,21,11, 4,20,13, 7,19, 8, 3,18,10, 6,17,12, 2,16,14, 5,15, 9)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,17,10)( 2,19,13)( 3,21, 9)( 4,16,12)( 5,18, 8)( 6,20,11)( 7,15,14)$
$ 21 $ $42$ $21$ $( 1,20,11, 7,16,13, 6,19, 8, 5,15,10, 4,18,12, 3,21,14, 2,17, 9)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,18,11)( 2,15, 9)( 3,19,14)( 4,16,12)( 5,20,10)( 6,17, 8)( 7,21,13)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 9,18)( 2, 8,19)( 3,14,20)( 4,13,21)( 5,12,15)( 6,11,16)( 7,10,17)$
$ 21 $ $42$ $21$ $( 1,10,19, 3,13,20, 5, 9,21, 7,12,15, 2, 8,16, 4,11,17, 6,14,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,12,19)( 2,10,16)( 3, 8,20)( 4,13,17)( 5,11,21)( 6, 9,18)( 7,14,15)$
$ 21 $ $42$ $21$ $( 1,12,15, 3,11,19, 5,10,16, 7, 9,20, 2, 8,17, 4,14,21, 6,13,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $7$ $3$ $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$ 6, 6, 6, 1, 1, 1 $ $49$ $6$ $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$
$ 6, 6, 6, 1, 1, 1 $ $49$ $6$ $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,18, 8, 2,17,14)( 3,16,13, 7,19, 9)( 4,15,12, 6,20,10)( 5,21,11)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,21,14)( 2,19,10, 7,16,11)( 3,17,13, 6,18, 8)( 4,15, 9, 5,20,12)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,20, 9, 2,16,14)( 3,19,12, 7,17,11)( 4,15,10, 6,21,13)( 5,18, 8)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1, 9,20, 7, 8,19)( 2,10,21, 6,14,18)( 3,11,15, 5,13,17)( 4,12,16)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,10,20, 6,13,19)( 2,12,17, 5,11,15)( 3,14,21, 4, 9,18)( 7, 8,16)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,12,19)( 2, 9,21, 7, 8,17)( 3,13,16, 6,11,15)( 4,10,18, 5,14,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $882=2 \cdot 3^{2} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  882.36
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);