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Properties

Label 21T17
21T17 1 4 1->4 12 1->12 2 3 2->3 13 2->13 14 3->14 8 4->8 5 7 5->7 9 5->9 6 10 6->10 11 7->11 18 8->18 8->18 17 9->17 19 9->19 16 10->16 20 10->20 15 11->15 21 11->21 12->15 12->21 13->16 13->20 14->17 14->19 15->7 15->9 16->6 16->10 17->5 17->11 18->4 18->12 19->3 19->13 20->2 20->14 21->1 21->8
Degree $21$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:S_3$

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Copy content magma:G := TransitiveGroup(21, 17);
 

Group invariants

Abstract group:  $C_7^2:S_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $294=2 \cdot 3 \cdot 7^{2}$
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 $t$:  $17$
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:  $(1,12,21)(2,13,20)(3,14,19)(4,8,18)(5,9,17)(6,10,16)(7,11,15)$, $(1,4)(2,3)(5,7)(8,18,12,15,9,19,13,16,10,20,14,17,11,21)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T15, 21T18, 42T56, 42T57, 42T62

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 $2^{10},1$ $21$ $2$ $10$ $( 1,18)( 2,19)( 3,20)( 4,21)( 5,15)( 6,16)( 7,17)( 9,14)(10,13)(11,12)$
3A $3^{7}$ $98$ $3$ $14$ $( 1,11,16)( 2,12,15)( 3,13,21)( 4,14,20)( 5, 8,19)( 6, 9,18)( 7,10,17)$
7A1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 2, 3, 4, 5, 6, 7)(15,16,17,18,19,20,21)$
7A-1 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,21,20,19,18,17,16)$
7A2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 3, 5, 7, 2, 4, 6)(15,17,19,21,16,18,20)$
7A-2 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 6, 4, 2, 7, 5, 3)(15,20,18,16,21,19,17)$
7A3 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)(15,18,21,17,20,16,19)$
7A-3 $7^{2},1^{7}$ $3$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,19,16,20,17,21,18)$
7B1 $7^{3}$ $6$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
7B-1 $7^{3}$ $6$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
7C1 $7^{3}$ $6$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
7C2 $7^{3}$ $6$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
7C3 $7^{3}$ $6$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
14A1 $14,2^{3},1$ $21$ $14$ $16$ $( 1,15, 2,16, 3,17, 4,18, 5,19, 6,20, 7,21)( 9,14)(10,13)(11,12)$
14A-1 $14,2^{3},1$ $21$ $14$ $16$ $( 1,21, 7,20, 6,19, 5,18, 4,17, 3,16, 2,15)( 9,14)(10,13)(11,12)$
14A3 $14,2^{3},1$ $21$ $14$ $16$ $( 1,16, 4,19, 7,15, 3,18, 6,21, 2,17, 5,20)( 9,14)(10,13)(11,12)$
14A-3 $14,2^{3},1$ $21$ $14$ $16$ $( 1,20, 5,17, 2,21, 6,18, 3,15, 7,19, 4,16)( 9,14)(10,13)(11,12)$
14A5 $14,2^{3},1$ $21$ $14$ $16$ $( 1,17, 6,15, 4,20, 2,18, 7,16, 5,21, 3,19)( 9,14)(10,13)(11,12)$
14A-5 $14,2^{3},1$ $21$ $14$ $16$ $( 1,19, 3,21, 5,16, 7,18, 2,20, 4,15, 6,17)( 9,14)(10,13)(11,12)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A 7A1 7A-1 7A2 7A-2 7A3 7A-3 7B1 7B-1 7C1 7C2 7C3 14A1 14A-1 14A3 14A-3 14A5 14A-5
Size 1 21 98 3 3 3 3 3 3 6 6 6 6 6 21 21 21 21 21 21
2 P 1A 1A 3A 7A2 7A-2 7A-3 7A3 7A-1 7A1 7B1 7B-1 7C2 7C3 7C1 7A1 7A-1 7A3 7A-3 7A-2 7A2
3 P 1A 2A 1A 7A3 7A-3 7A-1 7A1 7A2 7A-2 7B-1 7B1 7C3 7C1 7C2 14A3 14A-3 14A-5 14A5 14A1 14A-1
7 P 1A 2A 3A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
294.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.7.1b R 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
294.7.2a R 2 0 −1 2 2 2 2 2 2 2 2 2 2 2 0 0 0 0 0 0
294.7.3a1 C 3 1 0 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ72+ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ7−3+ζ7−1 ζ7−3+1+ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+ζ7+ζ72 ζ7 ζ72 ζ73 ζ7−1 ζ7−2 ζ7−3
294.7.3a2 C 3 1 0 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7−3+2ζ7−2 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7+2ζ73 ζ7−3+1+ζ73 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−1 ζ7−2 ζ7−3 ζ7 ζ72 ζ73
294.7.3a3 C 3 1 0 2ζ7−3+ζ7−1 ζ7−2+2ζ7 ζ7+2ζ73 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−1+ζ72 ζ7−1+1+ζ7 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−2 ζ73 ζ7 ζ72 ζ7−3 ζ7−1
294.7.3a4 C 3 1 0 ζ7+2ζ73 2ζ7−1+ζ72 2ζ7−3+ζ7−1 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−2+2ζ7 ζ7−1+1+ζ7 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+ζ7+ζ72 ζ72 ζ7−3 ζ7−1 ζ7−2 ζ73 ζ7
294.7.3a5 C 3 1 0 2ζ7−1+ζ72 ζ7−3+2ζ7−2 ζ7−2+2ζ7 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ72+ζ73 ζ7−2+1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−3+ζ7+ζ72 ζ7−3 ζ7 ζ7−2 ζ73 ζ7−1 ζ72
294.7.3a6 C 3 1 0 ζ7−2+2ζ7 2ζ72+ζ73 2ζ7−1+ζ72 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−3+2ζ7−2 ζ7−2+1+ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ7−3−1−ζ7−ζ72 ζ73 ζ7−1 ζ72 ζ7−3 ζ7 ζ7−2
294.7.3b1 C 3 −1 0 ζ7−3+2ζ7−2 ζ7+2ζ73 2ζ72+ζ73 2ζ7−1+ζ72 ζ7−2+2ζ7 2ζ7−3+ζ7−1 ζ7−3+1+ζ73 −ζ7−3−1−ζ7−ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 ζ7−3+ζ7+ζ72 −ζ7 −ζ72 −ζ73 −ζ7−1 −ζ7−2 −ζ7−3
294.7.3b2 C 3 −1 0 2ζ72+ζ73 2ζ7−3+ζ7−1 ζ7−3+2ζ7−2 ζ7−2+2ζ7 2ζ7−1+ζ72 ζ7+2ζ73 ζ7−3+1+ζ73 ζ7−3+ζ7+ζ72 ζ7−1+1+ζ7 ζ7−2+1+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−1 −ζ7−2 −ζ7−3 −ζ7 −ζ72 −ζ73
294.7.3b3 C 3 −1 0 2ζ7−3+ζ7−1 ζ7−2+2ζ7 ζ7+2ζ73 2ζ72+ζ73 ζ7−3+2ζ7−2 2ζ7−1+ζ72 ζ7−1+1+ζ7 ζ7−3+ζ7+ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 −ζ7−3−1−ζ7−ζ72 −ζ7−2 −ζ73 −ζ7 −ζ72 −ζ7−3 −ζ7−1
294.7.3b4 C 3 −1 0 ζ7+2ζ73 2ζ7−1+ζ72 2ζ7−3+ζ7−1 ζ7−3+2ζ7−2 2ζ72+ζ73 ζ7−2+2ζ7 ζ7−1+1+ζ7 −ζ7−3−1−ζ7−ζ72 ζ7−2+1+ζ72 ζ7−3+1+ζ73 ζ7−3+ζ7+ζ72 −ζ72 −ζ7−3 −ζ7−1 −ζ7−2 −ζ73 −ζ7
294.7.3b5 C 3 −1 0 2ζ7−1+ζ72 ζ7−3+2ζ7−2 ζ7−2+2ζ7 ζ7+2ζ73 2ζ7−3+ζ7−1 2ζ72+ζ73 ζ7−2+1+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 ζ7−3+ζ7+ζ72 −ζ7−3 −ζ7 −ζ7−2 −ζ73 −ζ7−1 −ζ72
294.7.3b6 C 3 −1 0 ζ7−2+2ζ7 2ζ72+ζ73 2ζ7−1+ζ72 2ζ7−3+ζ7−1 ζ7+2ζ73 ζ7−3+2ζ7−2 ζ7−2+1+ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ73 ζ7−1+1+ζ7 −ζ7−3−1−ζ7−ζ72 −ζ73 −ζ7−1 −ζ72 −ζ7−3 −ζ7 −ζ7−2
294.7.6a1 C 6 0 0 −2ζ7−3−2−2ζ7−2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 2ζ7−3+2ζ7+2ζ72 −1 ζ7−3+3+ζ7+ζ72 −1 −1 −ζ7−3+2−ζ7−ζ72 0 0 0 0 0 0
294.7.6a2 C 6 0 0 2ζ7−3+2ζ7+2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −2ζ7−3−2−2ζ7−2ζ72 −1 −ζ7−3+2−ζ7−ζ72 −1 −1 ζ7−3+3+ζ7+ζ72 0 0 0 0 0 0
294.7.6b1 R 6 0 0 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−3+2+2ζ73 2ζ7−2+2+2ζ72 2ζ7−2+2+2ζ72 2ζ7−1+2+2ζ7 ζ7−3+2ζ7−2+2ζ72+ζ73 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −1 0 0 0 0 0 0
294.7.6b2 R 6 0 0 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−2+2+2ζ72 2ζ7−1+2+2ζ7 2ζ7−1+2+2ζ7 2ζ7−3+2+2ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −1 0 0 0 0 0 0
294.7.6b3 R 6 0 0 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−1+2+2ζ7 2ζ7−3+2+2ζ73 2ζ7−3+2+2ζ73 2ζ7−2+2+2ζ72 ζ7−3−ζ7−2−1−ζ72+ζ73 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 −1 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed