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Properties

Label 21T38
21T38 1 7 1->7 2 3 2->3 8 2->8 9 3->9 4 5 4->5 10 4->10 6 5->6 11 5->11 6->1 6->4 7->8 12 7->12 13 8->13 9->10 14 9->14 10->11 15 10->15 11->2 11->9 16 12->16 17 13->17 13->17 18 14->18 14->18 15->3 15->16 16->14 19 16->19 17->15 20 17->20 18->4 18->13 21 19->21 19->21 20->5 20->19 21->6 21->20
Degree $21$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_7$

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Copy content magma:G := TransitiveGroup(21, 38);
 

Group invariants

Abstract group:  $S_7$
Copy content magma:IdentifyGroup(G);
 
Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $38$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15)$, $(2,3)(4,5,6)(7,8)(9,10,11)(13,17,15,16,14,18)(19,21,20)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 7: None

Low degree siblings

7T7, 14T46, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418

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

Malle's constant $a(G)$:     $1/5$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 7A 10A 12A
Size 1 21 105 105 70 280 210 630 504 210 420 840 720 504 420
2 P 1A 1A 1A 1A 3A 3B 2B 2B 5A 3A 3A 3B 7A 5A 6A
3 P 1A 2A 2B 2C 1A 1A 4A 4B 5A 2B 2A 2C 7A 10A 4A
5 P 1A 2A 2B 2C 3A 3B 4A 4B 1A 6A 6B 6C 7A 2A 12A
7 P 1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 1A 10A 12A
Type
5040.w.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5040.w.1b R 1 −1 1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1
5040.w.6a R 6 4 2 0 3 0 2 0 1 −1 1 0 −1 −1 −1
5040.w.6b R 6 −4 2 0 3 0 −2 0 1 −1 −1 0 −1 1 1
5040.w.14a R 14 6 2 2 2 −1 0 0 −1 2 0 −1 0 1 0
5040.w.14b R 14 −4 2 0 −1 2 2 0 −1 −1 −1 0 0 1 −1
5040.w.14c R 14 4 2 0 −1 2 −2 0 −1 −1 1 0 0 −1 1
5040.w.14d R 14 −6 2 −2 2 −1 0 0 −1 2 0 1 0 −1 0
5040.w.15a R 15 −5 −1 3 3 0 −1 −1 0 −1 1 0 1 0 −1
5040.w.15b R 15 5 −1 −3 3 0 1 −1 0 −1 −1 0 1 0 1
5040.w.20a R 20 0 −4 0 2 2 0 0 0 2 0 0 −1 0 0
5040.w.21a R 21 −1 1 3 −3 0 1 −1 1 1 −1 0 0 −1 1
5040.w.21b R 21 1 1 −3 −3 0 −1 −1 1 1 1 0 0 1 −1
5040.w.35a R 35 5 −1 1 −1 −1 −1 1 0 −1 −1 1 0 0 −1
5040.w.35b R 35 −5 −1 −1 −1 −1 1 1 0 −1 1 −1 0 0 1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed