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Properties

Label 21T21
21T21 1 2 1->2 8 1->8 4 2->4 13 2->13 3 6 3->6 11 3->11 4->1 9 4->9 5 5->3 14 5->14 6->5 12 6->12 7 10 7->10 8->12 21 8->21 9->14 19 9->19 10->9 17 10->17 15 11->15 12->13 20 12->20 13->8 18 13->18 14->10 16 14->16 15->2 15->18 16->4 16->20 17->6 17->15 18->1 18->17 19->3 20->5 20->21 21->7 21->16
Degree $21$
Order $441$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:C_3^2$

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

Group invariants

Abstract group:  $C_7^2:C_3^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $441=3^{2} \cdot 7^{2}$
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 $t$:  $21$
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:  $(1,2,4)(3,6,5)(8,12,13)(9,14,10)(15,18,17)(16,20,21)$, $(1,8,21,7,10,17,6,12,20,5,14,16,4,9,19,3,11,15,2,13,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$21$:  $C_7:C_3$ x 2
$63$:  21T7 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T21

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
3A1 $3^{7}$ $7$ $3$ $14$ $( 1, 9,16)( 2,14,20)( 3,12,17)( 4,10,21)( 5, 8,18)( 6,13,15)( 7,11,19)$
3A-1 $3^{7}$ $7$ $3$ $14$ $( 1,16, 9)( 2,20,14)( 3,17,12)( 4,21,10)( 5,18, 8)( 6,15,13)( 7,19,11)$
3B1 $3^{7}$ $7$ $3$ $14$ $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$
3B-1 $3^{7}$ $7$ $3$ $14$ $( 1,20,13)( 2,15, 9)( 3,17,12)( 4,19, 8)( 5,21,11)( 6,16,14)( 7,18,10)$
3C1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 6, 5)( 2, 3, 7)( 9,12,10)(11,13,14)(15,17,18)(16,21,20)$
3C-1 $3^{6},1^{3}$ $49$ $3$ $12$ $( 1, 5, 6)( 2, 7, 3)( 9,10,12)(11,14,13)(15,18,17)(16,20,21)$
3D1 $3^{7}$ $49$ $3$ $14$ $( 1,19,11)( 2,20,10)( 3,21, 9)( 4,15, 8)( 5,16,14)( 6,17,13)( 7,18,12)$
3D-1 $3^{7}$ $49$ $3$ $14$ $( 1,11,19)( 2,10,20)( 3, 9,21)( 4, 8,15)( 5,14,16)( 6,13,17)( 7,12,18)$
7A1 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
7A-1 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7B1 $7^{3}$ $3$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7B-1 $7^{3}$ $3$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
7C1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)$
7C-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)$
7D1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
7D-1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
21A1 $21$ $21$ $21$ $20$ $( 1,11,15, 5,10,17, 2, 9,19, 6, 8,21, 3,14,16, 7,13,18, 4,12,20)$
21A-1 $21$ $21$ $21$ $20$ $( 1,17, 8, 7,20,10, 6,16,12, 5,19,14, 4,15, 9, 3,18,11, 2,21,13)$
21A2 $21$ $21$ $21$ $20$ $( 1,15,10, 2,19, 8, 3,16,13, 4,20,11, 5,17, 9, 6,21,14, 7,18,12)$
21A-2 $21$ $21$ $21$ $20$ $( 1, 8,20, 6,12,19, 4, 9,18, 2,13,17, 7,10,16, 5,14,15, 3,11,21)$
21B1 $21$ $21$ $21$ $20$ $( 1,10,16, 5, 8,17, 2,13,18, 6,11,19, 3, 9,20, 7,14,21, 4,12,15)$
21B-1 $21$ $21$ $21$ $20$ $( 1,21, 9, 6,17,10, 4,20,11, 2,16,12, 7,19,13, 5,15,14, 3,18, 8)$
21B2 $21$ $21$ $21$ $20$ $( 1,18,14, 5,19,12, 2,20,10, 6,21, 8, 3,15,13, 7,16,11, 4,17, 9)$
21B-2 $21$ $21$ $21$ $20$ $( 1,11,15, 6,12,18, 4,13,21, 2,14,17, 7, 8,20, 5, 9,16, 3,10,19)$

Malle's constant $a(G)$:     $1/12$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 7A1 7A-1 7B1 7B-1 7C1 7C-1 7D1 7D-1 21A1 21A-1 21A2 21A-2 21B1 21B-1 21B2 21B-2
Size 1 7 7 7 7 49 49 49 49 3 3 3 3 9 9 9 9 21 21 21 21 21 21 21 21
3 P 1A 3A-1 3A1 3B-1 3B1 3C-1 3C1 3D-1 3D1 7A1 7A-1 7B1 7B-1 7C1 7C-1 7D1 7D-1 21A2 21A-2 21A1 21A-1 21B2 21B-2 21B1 21B-1
7 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 7A-1 7A1 7B-1 7B1 7C-1 7C1 7D-1 7D1 7A1 7A-1 7A1 7A-1 7B1 7B-1 7B1 7B-1
Type
441.9.1a 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
441.9.1b1 C 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3
441.9.1b2 C 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1c1 C 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1c2 C 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3
441.9.1d1 C 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
441.9.1d2 C 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
441.9.1e1 C 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1
441.9.1e2 C 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
441.9.3a1 C 3 0 0 3 3 0 0 0 0 −ζ7−3−1−ζ7−ζ72 3 ζ7−3+ζ7+ζ72 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
441.9.3a2 C 3 0 0 3 3 0 0 0 0 ζ7−3+ζ7+ζ72 3 −ζ7−3−1−ζ7−ζ72 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
441.9.3b1 C 3 3 3 0 0 0 0 0 0 3 −ζ7−3−1−ζ7−ζ72 3 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 0 0 0 0
441.9.3b2 C 3 3 3 0 0 0 0 0 0 3 ζ7−3+ζ7+ζ72 3 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 0 0 0 0
441.9.3c1 C 3 3ζ21−7 3ζ217 0 0 0 0 0 0 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 0 0 0 0
441.9.3c2 C 3 3ζ217 3ζ21−7 0 0 0 0 0 0 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 0 0 0 0
441.9.3c3 C 3 3ζ21−7 3ζ217 0 0 0 0 0 0 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 0 0 0 0
441.9.3c4 C 3 3ζ217 3ζ21−7 0 0 0 0 0 0 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 0 0 0 0
441.9.3d1 C 3 0 0 3ζ21−7 3ζ217 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218
441.9.3d2 C 3 0 0 3ζ217 3ζ21−7 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219
441.9.3d3 C 3 0 0 3ζ21−7 3ζ217 0 0 0 0 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 0 0 0 0 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−ζ212+ζ214−ζ219 −ζ21+ζ212−ζ214−ζ217+ζ219 ζ21−10+ζ212+ζ218
441.9.3d4 C 3 0 0 3ζ217 3ζ21−7 0 0 0 0 −ζ21−10−ζ21−ζ214−ζ218+ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 3 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 ζ21−10−1+ζ21+ζ214+ζ218−ζ219 −ζ21−10−ζ21−ζ214−ζ218+ζ219 0 0 0 0 ζ21−ζ212+ζ214−ζ219 −ζ21−10+1−ζ212+ζ217−ζ218 ζ21−10+ζ212+ζ218 −ζ21+ζ212−ζ214−ζ217+ζ219
441.9.9a1 C 9 0 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 −ζ7−3−2−ζ7−ζ72 2 2 ζ7−3−1+ζ7+ζ72 0 0 0 0 0 0 0 0
441.9.9a2 C 9 0 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 ζ7−3−1+ζ7+ζ72 2 2 −ζ7−3−2−ζ7−ζ72 0 0 0 0 0 0 0 0
441.9.9b1 C 9 0 0 0 0 0 0 0 0 −3ζ7−3−3−3ζ7−3ζ72 −3ζ7−3−3−3ζ7−3ζ72 3ζ7−3+3ζ7+3ζ72 3ζ7−3+3ζ7+3ζ72 2 ζ7−3−1+ζ7+ζ72 −ζ7−3−2−ζ7−ζ72 2 0 0 0 0 0 0 0 0
441.9.9b2 C 9 0 0 0 0 0 0 0 0 3ζ7−3+3ζ7+3ζ72 3ζ7−3+3ζ7+3ζ72 −3ζ7−3−3−3ζ7−3ζ72 −3ζ7−3−3−3ζ7−3ζ72 2 −ζ7−3−2−ζ7−ζ72 ζ7−3−1+ζ7+ζ72 2 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed