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Properties

Label 21T24
21T24 1 5 1->5 8 1->8 2 7 2->7 9 2->9 3 3->2 10 3->10 4 11 4->11 6 5->6 12 5->12 6->1 13 6->13 7->3 14 7->14 18 8->18 9->10 19 9->19 10->12 20 10->20 11->14 21 11->21 12->9 15 12->15 13->11 16 13->16 14->13 17 14->17 15->5 15->18 16->4 16->20 17->3 17->15 18->2 18->17 19->1 20->7 20->21 21->6 21->16
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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Copy content magma:G := TransitiveGroup(21, 24);
 

Group invariants

Abstract group:  $C_7:(C_3\times F_7)$
Copy content magma:IdentifyGroup(G);
 
Order:  $882=2 \cdot 3^{2} \cdot 7^{2}$
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 $t$:  $24$
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:  $(1,5,6)(2,7,3)(9,10,12)(11,14,13)(15,18,17)(16,20,21)$, $(1,8,18,2,9,19)(3,10,20,7,14,17)(4,11,21,6,13,16)(5,12,15)$
Copy content 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 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

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{3}$ $49$ $2$ $9$ $( 1, 2)( 3, 7)( 4, 6)( 9,14)(10,13)(11,12)(15,21)(16,20)(17,19)$
3A1 $3^{7}$ $7$ $3$ $14$ $( 1,20,13)( 2,17,11)( 3,21, 9)( 4,18,14)( 5,15,12)( 6,19,10)( 7,16, 8)$
3A-1 $3^{7}$ $7$ $3$ $14$ $( 1,13,20)( 2,11,17)( 3, 9,21)( 4,14,18)( 5,12,15)( 6,10,19)( 7, 8,16)$
3B1 $3^{7}$ $7$ $3$ $14$ $( 1, 9,16)( 2,12,18)( 3, 8,20)( 4,11,15)( 5,14,17)( 6,10,19)( 7,13,21)$
3B-1 $3^{7}$ $7$ $3$ $14$ $( 1,16, 9)( 2,18,12)( 3,20, 8)( 4,15,11)( 5,17,14)( 6,19,10)( 7,21,13)$
3C1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 3, 4)( 2, 7, 6)( 9,12,10)(11,13,14)(15,20,19)(16,17,21)$
3C-1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 4, 3)( 2, 6, 7)( 9,10,12)(11,14,13)(15,19,20)(16,21,17)$
3D1 $3^{7}$ $49$ $3$ $14$ $( 1,15,11)( 2,16,10)( 3,17, 9)( 4,18, 8)( 5,19,14)( 6,20,13)( 7,21,12)$
3D-1 $3^{7}$ $49$ $3$ $14$ $( 1,11,15)( 2,10,16)( 3, 9,17)( 4, 8,18)( 5,14,19)( 6,13,20)( 7,12,21)$
6A1 $6^{3},1^{3}$ $49$ $6$ $15$ $( 1, 6, 3, 2, 4, 7)( 9,13,12,14,10,11)(15,17,20,21,19,16)$
6A-1 $6^{3},1^{3}$ $49$ $6$ $15$ $( 1, 7, 4, 2, 3, 6)( 9,11,10,14,12,13)(15,16,19,21,20,17)$
6B1 $6^{3},3$ $49$ $6$ $17$ $( 1, 9,20, 3,13,21)( 2,11,17)( 4, 8,18, 7,14,16)( 5,10,15, 6,12,19)$
6B-1 $6^{3},3$ $49$ $6$ $17$ $( 1,21,13, 3,20, 9)( 2,17,11)( 4,16,14, 7,18, 8)( 5,19,12, 6,15,10)$
6C1 $6^{3},3$ $49$ $6$ $17$ $( 1, 9,15, 3,11,17)( 2,10,16)( 4,12,18, 7, 8,21)( 5,13,19, 6,14,20)$
6C-1 $6^{3},3$ $49$ $6$ $17$ $( 1,17,11, 3,15, 9)( 2,16,10)( 4,21, 8, 7,18,12)( 5,20,14, 6,19,13)$
6D1 $6^{3},3$ $49$ $6$ $17$ $( 1,19, 9, 6,16,10)( 2,17,12, 5,18,14)( 3,15, 8, 4,20,11)( 7,21,13)$
6D-1 $6^{3},3$ $49$ $6$ $17$ $( 1,10,16, 6, 9,19)( 2,14,18, 5,12,17)( 3,11,20, 4, 8,15)( 7,13,21)$
7A $7^{3}$ $6$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7B $7^{3}$ $6$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7C $7^{3}$ $18$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
7D $7^{2},1^{7}$ $18$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)$
21A1 $21$ $42$ $21$ $20$ $( 1,13,19, 5,11,20, 2, 9,21, 6,14,15, 3,12,16, 7,10,17, 4, 8,18)$
21A-1 $21$ $42$ $21$ $20$ $( 1,18, 8, 4,17,10, 7,16,12, 3,15,14, 6,21, 9, 2,20,11, 5,19,13)$
21B1 $21$ $42$ $21$ $20$ $( 1,10,20, 4,11,18, 7,12,16, 3,13,21, 6,14,19, 2, 8,17, 5, 9,15)$
21B-1 $21$ $42$ $21$ $20$ $( 1,15, 9, 5,17, 8, 2,19,14, 6,21,13, 3,16,12, 7,18,11, 4,20,10)$

Malle's constant $a(G)$:     $1/9$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 7A 7B 7C 7D 21A1 21A-1 21B1 21B-1
Size 1 49 7 7 7 7 49 49 49 49 49 49 49 49 49 49 49 49 6 6 18 18 42 42 42 42
2 P 1A 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 3C1 3C-1 3A1 3A-1 3D1 3D-1 3B1 3B-1 7A 7B 7C 7D 21A-1 21A1 21B-1 21B1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 7A 7B 7C 7D 7A 7A 7B 7B
7 P 1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 1A 1A 1A 1A 3B1 3B-1 3A-1 3A1
Type
882.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
882.36.1b R 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1 1 1
882.36.1c1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3 1 1 ζ3−1 ζ3−1 ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
882.36.1c2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3−1 1 1 ζ3 ζ3 ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
882.36.1d1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
882.36.1d2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 1 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1
882.36.1e1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 ζ3−1 ζ3 1 1
882.36.1e2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3 ζ3−1 1 1
882.36.1f1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3−1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1
882.36.1f2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3
882.36.1g1 C 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3 −1 −1 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
882.36.1g2 C 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3−1 −1 −1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
882.36.1h1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 −ζ3−1 −1 −ζ3 −ζ3−1 −ζ3 −1 −ζ3 −ζ3−1 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3
882.36.1h2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 −ζ3 −1 −ζ3−1 −ζ3 −ζ3−1 −1 −ζ3−1 −ζ3 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1
882.36.1i1 C 1 −1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 −1 −1 1 1 1 1 ζ3−1 ζ3 1 1
882.36.1i2 C 1 −1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 −1 −1 1 1 1 1 ζ3 ζ3−1 1 1
882.36.1j1 C 1 −1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 −1 −ζ3−1 −1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3 1 1 1 1 1 1 ζ3 ζ3−1
882.36.1j2 C 1 −1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 −1 −ζ3 −1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3
882.36.6a R 6 0 0 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 0 0 −1 −1
882.36.6b R 6 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −1 −1 0 0
882.36.6c1 C 6 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −ζ3−1 −ζ3 0 0
882.36.6c2 C 6 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 −1 −1 −1 −ζ3 −ζ3−1 0 0
882.36.6d1 C 6 0 0 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 0 0 −ζ3 −ζ3−1
882.36.6d2 C 6 0 0 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 −1 6 −1 −1 0 0 −ζ3−1 −ζ3
882.36.18a R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 −3 4 0 0 0 0
882.36.18b R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 4 −3 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed