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Properties

Label 21T43
Degree $21$
Order $6174$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:(C_3\times C_6)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 43);
 

Group invariants

Abstract group:  $C_7^3:(C_3\times C_6)$
magma: IdentifyGroup(G);
 
Order:  $6174=2 \cdot 3^{2} \cdot 7^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,17,4,14,15)(2,8,21,3,11,18)(5,10,19,7,9,20)(6,13,16)$, $(1,10,16,5,14,17,2,11,18,6,8,19,3,12,20,7,9,21,4,13,15)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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 3
$126$:  21T9 x 3
$882$:  21T24 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T43 x 11, 42T467 x 12

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{3}$ $343$ $2$ $9$ $( 2, 7)( 3, 6)( 4, 5)( 8,14)( 9,13)(10,12)(16,21)(17,20)(18,19)$
3A1 $3^{7}$ $49$ $3$ $14$ $( 1,15,11)( 2,19, 8)( 3,16,12)( 4,20, 9)( 5,17,13)( 6,21,10)( 7,18,14)$
3A-1 $3^{7}$ $49$ $3$ $14$ $( 1,11,15)( 2, 8,19)( 3,12,16)( 4, 9,20)( 5,13,17)( 6,10,21)( 7,14,18)$
3B1 $3^{7}$ $49$ $3$ $14$ $( 1,15,10)( 2,16,12)( 3,17,14)( 4,18, 9)( 5,19,11)( 6,20,13)( 7,21, 8)$
3B-1 $3^{7}$ $49$ $3$ $14$ $( 1,10,15)( 2,12,16)( 3,14,17)( 4, 9,18)( 5,11,19)( 6,13,20)( 7, 8,21)$
3C1 $3^{7}$ $49$ $3$ $14$ $( 1,16, 8)( 2,18, 9)( 3,20,10)( 4,15,11)( 5,17,12)( 6,19,13)( 7,21,14)$
3C-1 $3^{7}$ $49$ $3$ $14$ $( 1, 8,16)( 2, 9,18)( 3,10,20)( 4,11,15)( 5,12,17)( 6,13,19)( 7,14,21)$
3D1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 4, 3)( 2, 6, 7)( 8, 9,11)(10,13,12)(15,21,19)(17,18,20)$
3D-1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 3, 4)( 2, 7, 6)( 8,11, 9)(10,12,13)(15,19,21)(17,20,18)$
6A1 $6^{3},3$ $343$ $6$ $17$ $( 1,11,15)( 2,14,19, 7, 8,18)( 3,10,16, 6,12,21)( 4,13,20, 5, 9,17)$
6A-1 $6^{3},3$ $343$ $6$ $17$ $( 1,15,11)( 2,18, 8, 7,19,14)( 3,21,12, 6,16,10)( 4,17, 9, 5,20,13)$
6B1 $6^{3},3$ $343$ $6$ $17$ $( 1,12,20)( 2,11,15, 7,13,18)( 3,10,17, 6,14,16)( 4, 9,19, 5, 8,21)$
6B-1 $6^{3},3$ $343$ $6$ $17$ $( 1,20,12)( 2,18,13, 7,15,11)( 3,16,14, 6,17,10)( 4,21, 8, 5,19, 9)$
6C1 $6^{3},3$ $343$ $6$ $17$ $( 1,16,11, 3,21, 8)( 2,15,13)( 4,20,10, 7,17, 9)( 5,19,12, 6,18,14)$
6C-1 $6^{3},3$ $343$ $6$ $17$ $( 1, 8,21, 3,11,16)( 2,13,15)( 4, 9,17, 7,10,20)( 5,14,18, 6,12,19)$
6D1 $6^{3},1^{3}$ $343$ $6$ $15$ $( 1, 7, 4, 2, 3, 6)( 8,10, 9,13,11,12)(15,20,21,17,19,18)$
6D-1 $6^{3},1^{3}$ $343$ $6$ $15$ $( 1, 6, 3, 2, 4, 7)( 8,12,11,13, 9,10)(15,18,19,17,21,20)$
7A $7^{3}$ $6$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$
7B $7^{3}$ $6$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
7C $7^{3}$ $6$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
7D $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
7E $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)$
7F $7,1^{14}$ $18$ $7$ $6$ $(1,2,3,4,5,6,7)$
7G $7^{3}$ $18$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
7H $7^{3}$ $18$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
7I $7^{3}$ $18$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,18,21,17,20,16,19)$
7J $7^{3}$ $18$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
7K $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,21,20,19,18,17,16)$
7L $7^{2},1^{7}$ $18$ $7$ $12$ $( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7M $7^{2},1^{7}$ $18$ $7$ $12$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
7N $7^{3}$ $18$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7O $7^{3}$ $18$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
7P $7^{3}$ $18$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7Q $7^{3}$ $18$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
7R $7^{2},1^{7}$ $18$ $7$ $12$ $( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7S $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
7T $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,19,16,20,17,21,18)$
7U $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
21A1 $21$ $294$ $21$ $20$ $( 1,16,14, 4,19,13, 7,15,12, 3,18,11, 6,21,10, 2,17, 9, 5,20, 8)$
21A-1 $21$ $294$ $21$ $20$ $( 1, 8,20, 5, 9,17, 2,10,21, 6,11,18, 3,12,15, 7,13,19, 4,14,16)$
21B1 $21$ $294$ $21$ $20$ $( 1,15,14, 3,19, 9, 5,16,11, 7,20,13, 2,17, 8, 4,21,10, 6,18,12)$
21B-1 $21$ $294$ $21$ $20$ $( 1,12,18, 6,10,21, 4, 8,17, 2,13,20, 7,11,16, 5, 9,19, 3,14,15)$
21C1 $21$ $294$ $21$ $20$ $( 1, 8,19, 2,12,16, 3, 9,20, 4,13,17, 5,10,21, 6,14,18, 7,11,15)$
21C-1 $21$ $294$ $21$ $20$ $( 1,15,11, 7,18,14, 6,21,10, 5,17,13, 4,20, 9, 3,16,12, 2,19, 8)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

45 x 45 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed