↑

Properties

Label 42T30
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $D_7:A_4$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(42, 30);
 
Copy content sage:G = TransitiveGroup(42, 30)
 
Copy content oscar:G = transitive_group(42, 30)
 
Copy content gap:G := TransitiveGroup(42, 30);
 

Group invariants

Abstract group:  $D_7:A_4$
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:  $168=2^{3} \cdot 3 \cdot 7$
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 $t$:  $30$
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(1,23,37,15,36,8,29,2,24,38,16,35,7,30)(3,22,39,13,31,10,27)(4,21,40,14,32,9,28)(5,20,41,17,33,11,25,6,19,42,18,34,12,26)$, $(1,22,26,15,40,33)(2,21,25,16,39,34)(3,20,30,14,41,36)(4,19,29,13,42,35)(5,24,27,17,38,32)(6,23,28,18,37,31)(7,10,11,8,9,12)$
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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 7: $F_7$

Degree 14: None

Degree 21: 21T4

Low degree siblings

28T28, 42T31

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 $2^{14},1^{14}$ $3$ $2$ $14$ $( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)(25,26)(27,28)(31,32)(33,34)(39,40)(41,42)$
2B $2^{21}$ $7$ $2$ $21$ $( 1, 2)( 3, 4)( 5, 6)( 7,38)( 8,37)( 9,39)(10,40)(11,41)(12,42)(13,32)(14,31)(15,36)(16,35)(17,33)(18,34)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)$
2C $2^{19},1^{4}$ $21$ $2$ $19$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,19)( 6,20)( 7,15)( 8,16)( 9,14)(10,13)(11,17)(12,18)(25,41)(26,42)(27,39)(28,40)(29,38)(30,37)(35,36)$
3A1 $3^{14}$ $28$ $3$ $28$ $( 1, 3, 5)( 2, 4, 6)( 7,13,25)( 8,14,26)( 9,17,30)(10,18,29)(11,15,28)(12,16,27)(19,37,31)(20,38,32)(21,42,35)(22,41,36)(23,40,34)(24,39,33)$
3A-1 $3^{14}$ $28$ $3$ $28$ $( 1, 5, 3)( 2, 6, 4)( 7,25,13)( 8,26,14)( 9,30,17)(10,29,18)(11,28,15)(12,27,16)(19,31,37)(20,32,38)(21,35,42)(22,36,41)(23,34,40)(24,33,39)$
6A1 $6^{7}$ $28$ $6$ $35$ $( 1, 6, 3, 2, 5, 4)( 7,20,13,38,25,32)( 8,19,14,37,26,31)( 9,24,17,39,30,33)(10,23,18,40,29,34)(11,22,15,41,28,36)(12,21,16,42,27,35)$
6A-1 $6^{7}$ $28$ $6$ $35$ $( 1, 4, 5, 2, 3, 6)( 7,32,25,38,13,20)( 8,31,26,37,14,19)( 9,33,30,39,17,24)(10,34,29,40,18,23)(11,36,28,41,15,22)(12,35,27,42,16,21)$
7A $7^{6}$ $6$ $7$ $36$ $( 1, 7,16,24,29,36,37)( 2, 8,15,23,30,35,38)( 3,10,13,22,27,31,39)( 4, 9,14,21,28,32,40)( 5,12,18,19,25,33,41)( 6,11,17,20,26,34,42)$
14A1 $14^{2},7^{2}$ $6$ $14$ $38$ $( 1,29, 7,36,16,37,24)( 2,30, 8,35,15,38,23)( 3,28,10,32,13,40,22, 4,27, 9,31,14,39,21)( 5,26,12,34,18,42,19, 6,25,11,33,17,41,20)$
14A3 $14^{2},7^{2}$ $6$ $14$ $38$ $( 1,36,24, 7,37,29,16)( 2,35,23, 8,38,30,15)( 3,32,22, 9,39,28,13, 4,31,21,10,40,27,14)( 5,34,19,11,41,26,18, 6,33,20,12,42,25,17)$
14A5 $14^{2},7^{2}$ $6$ $14$ $38$ $( 1,30, 7,35,16,38,24, 2,29, 8,36,15,37,23)( 3,28,10,32,13,40,22, 4,27, 9,31,14,39,21)( 5,25,12,33,18,41,19)( 6,26,11,34,17,42,20)$

Malle's constant $a(G)$:     $1/14$

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 2B 2C 3A1 3A-1 6A1 6A-1 7A 14A1 14A3 14A5
Size 1 3 7 21 28 28 28 28 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 7A 7A 7A 7A
3 P 1A 2A 2B 2C 1A 1A 2B 2B 7A 14A3 14A5 14A1
7 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 1A 2A 2A 2A
Type
168.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1
168.49.1b R 1 1 −1 −1 1 1 −1 −1 1 1 1 1
168.49.1c1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1
168.49.1c2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1
168.49.1d1 C 1 1 −1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1
168.49.1d2 C 1 1 −1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1
168.49.3a R 3 −1 3 −1 0 0 0 0 3 −1 −1 −1
168.49.3b R 3 −1 −3 1 0 0 0 0 3 −1 −1 −1
168.49.6a R 6 6 0 0 0 0 0 0 −1 −1 −1 −1
168.49.6b1 R 6 −2 0 0 0 0 0 0 −1 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7
168.49.6b2 R 6 −2 0 0 0 0 0 0 −1 2ζ7−2+1+2ζ72 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73
168.49.6b3 R 6 −2 0 0 0 0 0 0 −1 2ζ7−1+1+2ζ7 2ζ7−3+1+2ζ73 2ζ7−2+1+2ζ72

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