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Properties

Label 21T48
21T48 1 6 1->6 15 1->15 2 3 2->3 21 2->21 7 3->7 20 3->20 4 19 4->19 5 5->1 18 5->18 6->5 17 6->17 7->2 16 7->16 8 8->1 12 8->12 9 9->3 14 9->14 10 10->5 10->9 11 11->7 12->2 13 12->13 13->4 13->8 14->6 14->10 15->9 16->12 17->8 18->11 19->14 20->10 21->13
Degree $21$
Order $9261$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:\He_3$

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Copy content magma:G := TransitiveGroup(21, 48);
 

Group invariants

Abstract group:  $C_7^3:\He_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $9261=3^{3} \cdot 7^{3}$
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 $t$:  $48$
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:  $(1,6,5)(2,3,7)(8,12,13)(9,14,10)$, $(1,15,9,3,20,10,5,18,11,7,16,12,2,21,13,4,19,14,6,17,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T48 x 3

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 $3^{7}$ $147$ $3$ $14$ $( 1,18, 8)( 2,17,10)( 3,16,12)( 4,15,14)( 5,21, 9)( 6,20,11)( 7,19,13)$
3A-1 $3^{7}$ $147$ $3$ $14$ $( 1, 8,18)( 2,10,17)( 3,12,16)( 4,14,15)( 5, 9,21)( 6,11,20)( 7,13,19)$
3B1 $3^{7}$ $147$ $3$ $14$ $( 1,17, 8)( 2,15,10)( 3,20,12)( 4,18,14)( 5,16, 9)( 6,21,11)( 7,19,13)$
3B-1 $3^{7}$ $147$ $3$ $14$ $( 1, 8,17)( 2,10,15)( 3,12,20)( 4,14,18)( 5, 9,16)( 6,11,21)( 7,13,19)$
3C1 $3^{7}$ $147$ $3$ $14$ $( 1,15,11)( 2,21,12)( 3,20,13)( 4,19,14)( 5,18, 8)( 6,17, 9)( 7,16,10)$
3C-1 $3^{7}$ $147$ $3$ $14$ $( 1,11,15)( 2,12,21)( 3,13,20)( 4,14,19)( 5, 8,18)( 6, 9,17)( 7,10,16)$
3D1 $3^{4},1^{9}$ $147$ $3$ $8$ $( 2, 5, 3)( 4, 6, 7)(15,18,17)(16,20,21)$
3D-1 $3^{4},1^{9}$ $147$ $3$ $8$ $( 2, 3, 5)( 4, 7, 6)(15,17,18)(16,21,20)$
3E1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 5, 6)( 2, 7, 3)( 8, 9,11)(10,13,12)(15,21,19)(17,18,20)$
3E-1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 6, 5)( 2, 3, 7)( 8,11, 9)(10,12,13)(15,19,21)(17,20,18)$
7A1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
7A-1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
7B1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
7B-1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
7C1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
7C-1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
7D1 $7,1^{14}$ $9$ $7$ $6$ $( 8,12, 9,13,10,14,11)$
7D-1 $7,1^{14}$ $9$ $7$ $6$ $( 8,11,14,10,13, 9,12)$
7E1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)$
7E-1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,21,20,19,18,17,16)$
7F1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,21,20,19,18,17,16)$
7F-1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7G1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7G-1 $7^{3}$ $27$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7H1 $7^{3}$ $27$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7H-1 $7^{3}$ $27$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7I1 $7^{3}$ $27$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7I-1 $7^{3}$ $27$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
21A1 $21$ $441$ $21$ $20$ $( 1,19,11, 5,15,12, 2,18,13, 6,21,14, 3,17, 8, 7,20, 9, 4,16,10)$
21A-1 $21$ $441$ $21$ $20$ $( 1,10,16, 4, 9,20, 7, 8,17, 3,14,21, 6,13,18, 2,12,15, 5,11,19)$
21A2 $21$ $441$ $21$ $20$ $( 1,11,15, 2,13,21, 3, 8,20, 4,10,19, 5,12,18, 6,14,17, 7, 9,16)$
21A-2 $21$ $441$ $21$ $20$ $( 1,16, 9, 7,17,14, 6,18,12, 5,19,10, 4,20, 8, 3,21,13, 2,15,11)$
21B1 $21$ $441$ $21$ $20$ $( 1,18,13, 3,21,10, 5,17,14, 7,20,11, 2,16, 8, 4,19,12, 6,15, 9)$
21B-1 $21$ $441$ $21$ $20$ $( 1, 9,15, 6,12,19, 4, 8,16, 2,11,20, 7,14,17, 5,10,21, 3,13,18)$
21B2 $21$ $441$ $21$ $20$ $( 1,13,21, 5,14,20, 2, 8,19, 6, 9,18, 3,10,17, 7,11,16, 4,12,15)$
21B-2 $21$ $441$ $21$ $20$ $( 1,15,12, 4,16,11, 7,17,10, 3,18, 9, 6,19, 8, 2,20,14, 5,21,13)$
21C1 $21$ $441$ $21$ $20$ $( 1,21,13, 4,18, 9, 7,15,12, 3,19, 8, 6,16,11, 2,20,14, 5,17,10)$
21C-1 $21$ $441$ $21$ $20$ $( 1,10,17, 5,14,20, 2,11,16, 6, 8,19, 3,12,15, 7, 9,18, 4,13,21)$
21C2 $21$ $441$ $21$ $20$ $( 1,13,18, 7,12,19, 6,11,20, 5,10,21, 4, 9,15, 3, 8,16, 2,14,17)$
21C-2 $21$ $441$ $21$ $20$ $( 1,17,14, 2,16, 8, 3,15, 9, 4,21,10, 5,20,11, 6,19,12, 7,18,13)$
21D1 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 2, 5, 3)( 4, 6, 7)( 8,14,13,12,11,10, 9)(15,18,17)(16,20,21)$
21D-1 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 2, 3, 5)( 4, 7, 6)( 8, 9,10,11,12,13,14)(15,17,18)(16,21,20)$
21D2 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 2, 3, 5)( 4, 7, 6)( 8,13,11, 9,14,12,10)(15,17,18)(16,21,20)$
21D-2 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 2, 5, 3)( 4, 6, 7)( 8,10,12,14, 9,11,13)(15,18,17)(16,20,21)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

45 x 45 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed