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Properties

Label 7T3
7T3 1 2 1->2 1->2 3 2->3 4 2->4 3->4 6 3->6 4->1 5 4->5 5->3 5->6 6->5 7 6->7 7->1
Degree $7$
Order $21$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_7:C_3$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(7, 3);
 

Group invariants

Abstract group:  $C_7:C_3$
Copy content magma:IdentifyGroup(G);
 
Order:  $21=3 \cdot 7$
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 $n$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $F_{21}(7) = 7:3$
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:  $(1,2,4)(3,6,5)$, $(1,2,3,4,5,6,7)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

21T2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{7}$ $1$ $1$ $0$ $()$
3A1 $3^{2},1$ $7$ $3$ $4$ $(1,4,3)(2,6,7)$
3A-1 $3^{2},1$ $7$ $3$ $4$ $(1,3,4)(2,7,6)$
7A1 $7$ $3$ $7$ $6$ $(1,2,3,4,5,6,7)$
7A-1 $7$ $3$ $7$ $6$ $(1,4,7,3,6,2,5)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 3A1 3A-1 7A1 7A-1
Size 1 7 7 3 3
3 P 1A 3A-1 3A1 7A1 7A-1
7 P 1A 1A 1A 7A-1 7A1
Type
21.1.1a R 1 1 1 1 1
21.1.1b1 C 1 ζ3−1 ζ3 1 1
21.1.1b2 C 1 ζ3 ζ3−1 1 1
21.1.3a1 C 3 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
21.1.3a2 C 3 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $64 x^{7}-112 \left(s^{2}+7 t^{2}\right) x^{5}+56 \left(s^{2}+7 t^{2}\right)^{2} x^{3}-7 \left(s^{2}+7 t^{2}\right)^{3} x-s \left(s^{2}+7 t^{2}\right)^{3}$ Copy content Toggle raw display
$f_{ 2 } =$ $27 x^{7} + \left(81 t^{2} + 3699\right) x^{6} + \left(-1377 t^{4} - 5670 t^{2} - 23328 t + 132759\right) x^{5} + \left(-5811 t^{6} - 153387 t^{4} - 113184 t^{3} - 966897 t^{2} - 1531872 t + 47007\right) x^{4} + \left(-1887 t^{8} - 195636 t^{6} - 46656 t^{5} - 3937194 t^{4} - 3450816 t^{3} - 25169940 t^{2} - 33203520 t - 48370959\right) x^{3} + \left(11715 t^{10} + 338733 t^{8} + 388800 t^{7} + 2731662 t^{6} + 7216128 t^{5} + 3300426 t^{4} + 22353408 t^{3} - 35833185 t^{2} - 82067904 t - 179039079\right) x^{2} + \left(8933 t^{12} + 412602 t^{10} + 360288 t^{9} + 7773795 t^{8} + 11772864 t^{7} + 82148172 t^{6} + 141787584 t^{5} + 538518267 t^{4} + 792220608 t^{3} + 2068073370 t^{2} + 1846590048 t + 3798843165\right) x + \left(-289 t^{14} - 22805 t^{12} + 14688 t^{11} - 710733 t^{10} + 315360 t^{9} - 11528625 t^{8} + 51840 t^{7} - 113796915 t^{6} - 39097728 t^{5} - 737041167 t^{4} - 321308640 t^{3} - 2901892959 t^{2} - 869800032 t - 5291543403\right)$ Copy content Toggle raw display