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Properties

Label 42T2
Degree $42$
Order $42$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_7:C_6$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(42, 2);
 
Copy content sage:G = TransitiveGroup(42, 2)
 
Copy content oscar:G = transitive_group(42, 2)
 
Copy content gap:G := TransitiveGroup(42, 2);
 

Group invariants

Abstract group:  $C_7:C_6$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $42=2 \cdot 3 \cdot 7$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  yes
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $42$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $2$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  1
Primitive:  no
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $42$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(42).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(42), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(42), G));
 
Generators:  $(1,18,24,2,17,23)(3,14,19,4,13,20)(5,16,21,6,15,22)(7,41,32,8,42,31)(9,37,33,10,38,34)(11,40,35,12,39,36)(25,30,27,26,29,28)$, $(1,7,13,22,28,36,38,2,8,14,21,27,35,37)(3,10,15,23,30,32,39,4,9,16,24,29,31,40)(5,12,17,20,26,34,41,6,11,18,19,25,33,42)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$21$:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Degree 21: 21T2

Low degree siblings

14T5

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{21}$ $1$ $2$ $21$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)$
3A1 $3^{14}$ $7$ $3$ $28$ $( 1, 3, 5)( 2, 4, 6)( 7,16,25)( 8,15,26)( 9,17,28)(10,18,27)(11,13,30)(12,14,29)(19,38,31)(20,37,32)(21,39,33)(22,40,34)(23,42,36)(24,41,35)$
3A-1 $3^{14}$ $7$ $3$ $28$ $( 1, 5, 3)( 2, 6, 4)( 7,25,16)( 8,26,15)( 9,28,17)(10,27,18)(11,30,13)(12,29,14)(19,31,38)(20,32,37)(21,33,39)(22,34,40)(23,36,42)(24,35,41)$
6A1 $6^{7}$ $7$ $6$ $35$ $( 1, 6, 3, 2, 5, 4)( 7,26,16, 8,25,15)( 9,27,17,10,28,18)(11,29,13,12,30,14)(19,32,38,20,31,37)(21,34,39,22,33,40)(23,35,42,24,36,41)$
6A-1 $6^{7}$ $7$ $6$ $35$ $( 1, 4, 5, 2, 3, 6)( 7,15,25, 8,16,26)( 9,18,28,10,17,27)(11,14,30,12,13,29)(19,37,31,20,38,32)(21,40,33,22,39,34)(23,41,36,24,42,35)$
7A1 $7^{6}$ $3$ $7$ $36$ $( 1,13,28,38, 8,21,35)( 2,14,27,37, 7,22,36)( 3,15,30,39, 9,24,31)( 4,16,29,40,10,23,32)( 5,17,26,41,11,19,33)( 6,18,25,42,12,20,34)$
7A-1 $7^{6}$ $3$ $7$ $36$ $( 1,35,21, 8,38,28,13)( 2,36,22, 7,37,27,14)( 3,31,24, 9,39,30,15)( 4,32,23,10,40,29,16)( 5,33,19,11,41,26,17)( 6,34,20,12,42,25,18)$
14A1 $14^{3}$ $3$ $14$ $39$ $( 1, 7,13,22,28,36,38, 2, 8,14,21,27,35,37)( 3,10,15,23,30,32,39, 4, 9,16,24,29,31,40)( 5,12,17,20,26,34,41, 6,11,18,19,25,33,42)$
14A-1 $14^{3}$ $3$ $14$ $39$ $( 1,22,38,14,35, 7,28, 2,21,37,13,36, 8,27)( 3,23,39,16,31,10,30, 4,24,40,15,32, 9,29)( 5,20,41,18,33,12,26, 6,19,42,17,34,11,25)$

Malle's constant $a(G)$:     $1/21$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 6A1 6A-1 7A1 7A-1 14A1 14A-1
Size 1 1 7 7 7 7 3 3 3 3
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A1 7A-1 7A1 7A-1
3 P 1A 2A 1A 1A 2A 2A 7A-1 7A1 14A-1 14A1
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 2A 2A
Type
42.2.1a R 1 1 1 1 1 1 1 1 1 1
42.2.1b R 1 −1 1 1 −1 −1 1 1 −1 −1
42.2.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
42.2.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
42.2.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 −1 −1
42.2.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 −1 −1
42.2.3a1 C 3 3 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
42.2.3a2 C 3 3 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
42.2.3b1 C 3 −3 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72
42.2.3b2 C 3 −3 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

Data not computed