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Properties

Label 21T42
21T42 1 2 1->2 14 1->14 5 2->5 13 2->13 3 3->1 12 3->12 4 11 4->11 7 5->7 10 5->10 6 6->3 9 6->9 7->6 8 7->8 8->9 21 8->21 9->12 17 9->17 10->8 20 10->20 16 11->16 12->14 19 12->19 13->10 15 13->15 14->13 18 14->18 15->2 15->20 16->1 17->7 17->19 18->6 18->15 19->5 19->18 20->4 20->21 21->3 21->17
Degree $21$
Order $6174$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:C_{18}$

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Copy content magma:G := TransitiveGroup(21, 42);
 

Group invariants

Abstract group:  $C_7^3:C_{18}$
Copy content magma:IdentifyGroup(G);
 
Order:  $6174=2 \cdot 3^{2} \cdot 7^{3}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,5,7,6,3)(8,9,12,14,13,10)(15,20,21,17,19,18)$, $(1,14,18,6,9,17,7,8,21,3,12,19,5,10,20,4,11,16)(2,13,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$18$:  $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T42 x 18, 42T466 x 19

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 $2^{9},1^{3}$ $343$ $2$ $9$ $( 1, 7)( 2, 6)( 3, 5)( 8,14)( 9,13)(10,12)(15,19)(16,18)(20,21)$
3A1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 6, 5)( 2, 3, 7)( 8,13,12)( 9,10,14)(15,16,20)(18,21,19)$
3A-1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 5, 6)( 2, 7, 3)( 8,12,13)( 9,14,10)(15,20,16)(18,19,21)$
6A1 $6^{3},1^{3}$ $343$ $6$ $15$ $( 1, 3, 6, 7, 5, 2)( 8,10,13,14,12, 9)(15,21,16,19,20,18)$
6A-1 $6^{3},1^{3}$ $343$ $6$ $15$ $( 1, 2, 5, 7, 6, 3)( 8, 9,12,14,13,10)(15,18,20,19,16,21)$
7A $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7B $7^{3}$ $18$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7C $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)(15,20,18,16,21,19,17)$
7D $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)$
7E $7^{3}$ $18$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$
7F $7^{3}$ $18$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
7G $7^{2},1^{7}$ $18$ $7$ $12$ $( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
7H $7^{3}$ $18$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
7I $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)$
7J $7^{3}$ $18$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
7K $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
7L $7^{3}$ $18$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7M $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)$
7N $7^{3}$ $18$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
7O $7^{3}$ $18$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7P $7^{3}$ $18$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
7Q $7^{3}$ $18$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
7R $7,1^{14}$ $18$ $7$ $6$ $(15,21,20,19,18,17,16)$
7S $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 6, 4, 2, 7, 5, 3)(15,17,19,21,16,18,20)$
9A1 $9^{2},3$ $343$ $9$ $18$ $( 1,13,21, 6,12,19, 5, 8,18)( 2,10,15, 3,14,16, 7, 9,20)( 4,11,17)$
9A-1 $9^{2},3$ $343$ $9$ $18$ $( 1,18, 8, 5,19,12, 6,21,13)( 2,20, 9, 7,16,14, 3,15,10)( 4,17,11)$
9A2 $9^{2},3$ $343$ $9$ $18$ $( 1,21,12, 5,18,13, 6,19, 8)( 2,15,14, 7,20,10, 3,16, 9)( 4,17,11)$
9A-2 $9^{2},3$ $343$ $9$ $18$ $( 1, 8,19, 6,13,18, 5,12,21)( 2, 9,16, 3,10,20, 7,14,15)( 4,11,17)$
9A4 $9^{2},3$ $343$ $9$ $18$ $( 1,12,18, 6, 8,21, 5,13,19)( 2,14,20, 3, 9,15, 7,10,16)( 4,11,17)$
9A-4 $9^{2},3$ $343$ $9$ $18$ $( 1,19,13, 5,21, 8, 6,18,12)( 2,16,10, 7,15, 9, 3,20,14)( 4,17,11)$
18A1 $18,3$ $343$ $18$ $19$ $( 1,15,13, 3,21,14, 6,16,12, 7,19, 9, 5,20, 8, 2,18,10)( 4,17,11)$
18A-1 $18,3$ $343$ $18$ $19$ $( 1,10,18, 2, 8,20, 5, 9,19, 7,12,16, 6,14,21, 3,13,15)( 4,11,17)$
18A5 $18,3$ $343$ $18$ $19$ $( 1,14,19, 2,13,16, 5,10,21, 7, 8,15, 6, 9,18, 3,12,20)( 4,11,17)$
18A-5 $18,3$ $343$ $18$ $19$ $( 1,20,12, 3,18, 9, 6,15, 8, 7,21,10, 5,16,13, 2,19,14)( 4,17,11)$
18A7 $18,3$ $343$ $18$ $19$ $( 1,16, 8, 3,19,10, 6,20,13, 7,18,14, 5,15,12, 2,21, 9)( 4,17,11)$
18A-7 $18,3$ $343$ $18$ $19$ $( 1, 9,21, 2,12,15, 5,14,18, 7,13,20, 6,10,19, 3, 8,16)( 4,11,17)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

37 x 37 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed