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Properties

Label 28T6
28T6 1 10 1->10 24 1->24 2 9 2->9 23 2->23 3 12 3->12 22 3->22 4 11 4->11 21 4->21 5 13 5->13 19 5->19 6 14 6->14 20 6->20 7 7->2 17 7->17 8 8->1 18 8->18 9->4 16 9->16 10->3 15 10->15 11->6 28 11->28 12->5 27 12->27 13->8 26 13->26 14->7 25 14->25 15->23 16->24 17->26 18->25 19->28 20->27 21->15 22->16 23->18 24->17 25->19 26->20 27->21 28->22
Degree $28$
Order $56$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_7:D_4$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(28, 6);
 
Copy content sage:G = TransitiveGroup(28, 6)
 
Copy content oscar:G = transitive_group(28, 6)
 
Copy content gap:G := TransitiveGroup(28, 6);
 

Group invariants

Abstract group:  $C_7:D_4$
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:  $56=2^{3} \cdot 7$
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 $n$:  $28$
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 $t$:  $6$
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);
 
$\card{\Aut(F/K)}$:  $14$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(28).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(28), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(28), G));
 
Generators:  $(1,24)(2,23)(3,22)(4,21)(5,19)(6,20)(7,17)(8,18)(9,16)(10,15)(11,28)(12,27)(13,26)(14,25)$, $(1,10,3,12,5,13,8)(2,9,4,11,6,14,7)(15,23,18,25,19,28,22,16,24,17,26,20,27,21)$
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)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$14$:  $D_{7}$
$28$:  $D_{14}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Low degree siblings

28T7

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{14}$ $1$ $2$ $14$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)$
2B $2^{7},1^{14}$ $2$ $2$ $7$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)$
2C $2^{14}$ $14$ $2$ $14$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15)(10,16)(11,27)(12,28)(13,25)(14,26)$
4A $4^{7}$ $14$ $4$ $21$ $( 1,16, 2,15)( 3,28, 4,27)( 5,25, 6,26)( 7,24, 8,23)( 9,22,10,21)(11,19,12,20)(13,17,14,18)$
7A1 $7^{4}$ $2$ $7$ $24$ $( 1, 3, 5, 8,10,12,13)( 2, 4, 6, 7, 9,11,14)(15,18,19,22,24,26,27)(16,17,20,21,23,25,28)$
7A2 $7^{4}$ $2$ $7$ $24$ $( 1, 5,10,13, 3, 8,12)( 2, 6, 9,14, 4, 7,11)(15,19,24,27,18,22,26)(16,20,23,28,17,21,25)$
7A3 $7^{4}$ $2$ $7$ $24$ $( 1, 8,13, 5,12, 3,10)( 2, 7,14, 6,11, 4, 9)(15,22,27,19,26,18,24)(16,21,28,20,25,17,23)$
14A1 $14^{2}$ $2$ $14$ $26$ $( 1, 9, 3,11, 5,14, 8, 2,10, 4,12, 6,13, 7)(15,23,18,25,19,28,22,16,24,17,26,20,27,21)$
14A3 $14^{2}$ $2$ $14$ $26$ $( 1, 6,10,14, 3, 7,12, 2, 5, 9,13, 4, 8,11)(15,20,24,28,18,21,26,16,19,23,27,17,22,25)$
14A5 $14^{2}$ $2$ $14$ $26$ $( 1,14,12, 9, 8, 6, 3, 2,13,11,10, 7, 5, 4)(15,28,26,23,22,20,18,16,27,25,24,21,19,17)$
14B1 $14,7^{2}$ $2$ $14$ $25$ $( 1, 9, 3,11, 5,14, 8, 2,10, 4,12, 6,13, 7)(15,24,18,26,19,27,22)(16,23,17,25,20,28,21)$
14B-1 $14,7^{2}$ $2$ $14$ $25$ $( 1, 7,13, 6,12, 4,10, 2, 8,14, 5,11, 3, 9)(15,22,27,19,26,18,24)(16,21,28,20,25,17,23)$
14B3 $14,7^{2}$ $2$ $14$ $25$ $( 1,11, 8, 4,13, 9, 5, 2,12, 7, 3,14,10, 6)(15,26,22,18,27,24,19)(16,25,21,17,28,23,20)$
14B-3 $14,7^{2}$ $2$ $14$ $25$ $( 1, 6,10,14, 3, 7,12, 2, 5, 9,13, 4, 8,11)(15,19,24,27,18,22,26)(16,20,23,28,17,21,25)$
14B5 $14,7^{2}$ $2$ $14$ $25$ $( 1,14,12, 9, 8, 6, 3, 2,13,11,10, 7, 5, 4)(15,27,26,24,22,19,18)(16,28,25,23,21,20,17)$
14B-5 $14,7^{2}$ $2$ $14$ $25$ $( 1, 4, 5, 7,10,11,13, 2, 3, 6, 8, 9,12,14)(15,18,19,22,24,26,27)(16,17,20,21,23,25,28)$

Malle's constant $a(G)$:     $1/7$

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 4A 7A1 7A2 7A3 14A1 14A3 14A5 14B1 14B-1 14B3 14B-3 14B5 14B-5
Size 1 1 2 14 14 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 7A2 7A3 7A1 7A1 7A3 7A2 7A1 7A1 7A3 7A3 7A2 7A2
7 P 1A 2A 2B 2C 4A 7A3 7A1 7A2 14A3 14A5 14A1 14B3 14B-3 14B-5 14B5 14B1 14B-1
Type
56.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
56.7.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.7.1c R 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
56.7.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
56.7.2a R 2 −2 0 0 0 2 2 2 −2 −2 −2 0 0 0 0 0 0
56.7.2b1 R 2 2 2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73
56.7.2b2 R 2 2 2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72
56.7.2b3 R 2 2 2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−1+ζ7
56.7.2c1 R 2 2 −2 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−2+ζ72 ζ7−1+ζ7 ζ7−3+ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73
56.7.2c2 R 2 2 −2 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−1+ζ7 ζ7−3+ζ73 ζ7−2+ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72
56.7.2c3 R 2 2 −2 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−3+ζ73 ζ7−2+ζ72 ζ7−1+ζ7 −ζ7−3−ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−1−ζ7
56.7.2d1 C 2 −2 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 ζ7−2−ζ72 −ζ7−2+ζ72 −ζ7−1+ζ7 ζ7−1−ζ7 ζ7−3−ζ73 −ζ7−3+ζ73
56.7.2d2 C 2 −2 0 0 0 ζ7−3+ζ73 ζ7−1+ζ7 ζ7−2+ζ72 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2+ζ72 ζ7−2−ζ72 ζ7−1−ζ7 −ζ7−1+ζ7 −ζ7−3+ζ73 ζ7−3−ζ73
56.7.2d3 C 2 −2 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 ζ7−1−ζ7 −ζ7−1+ζ7 ζ7−3−ζ73 −ζ7−3+ζ73 −ζ7−2+ζ72 ζ7−2−ζ72
56.7.2d4 C 2 −2 0 0 0 ζ7−2+ζ72 ζ7−3+ζ73 ζ7−1+ζ7 −ζ7−1−ζ7 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1+ζ7 ζ7−1−ζ7 −ζ7−3+ζ73 ζ7−3−ζ73 ζ7−2−ζ72 −ζ7−2+ζ72
56.7.2d5 C 2 −2 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 ζ7−3−ζ73 −ζ7−3+ζ73 ζ7−2−ζ72 −ζ7−2+ζ72 ζ7−1−ζ7 −ζ7−1+ζ7
56.7.2d6 C 2 −2 0 0 0 ζ7−1+ζ7 ζ7−2+ζ72 ζ7−3+ζ73 −ζ7−3−ζ73 −ζ7−2−ζ72 −ζ7−1−ζ7 −ζ7−3+ζ73 ζ7−3−ζ73 −ζ7−2+ζ72 ζ7−2−ζ72 −ζ7−1+ζ7 ζ7−1−ζ7

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