Properties

Label 36T939
Degree $36$
Order $576$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6^2:\SD_{16}$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 939);
 
Copy content sage:G = TransitiveGroup(36, 939)
 
Copy content oscar:G = transitive_group(36, 939)
 
Copy content gap:G := TransitiveGroup(36, 939);
 

Group invariants

Abstract group:  $C_6^2:\SD_{16}$
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:  $576=2^{6} \cdot 3^{2}$
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 $n$:  $36$
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 $t$:  $939$
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(36).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(36), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(36), G));
 
Generators:  $(1,28,11,32,23,36,15,5)(2,27,12,31,24,35,16,6)(3,25,9,30,21,33,14,7)(4,26,10,29,22,34,13,8)(17,20,18,19)$, $(1,8,12,16,29,23)(2,7,11,15,30,24)(3,6,10,14,32,22)(4,5,9,13,31,21)(17,33,25)(18,34,26)(19,36,27,20,35,28)$, $(1,17,33,8,29,16,23,25)(2,18,34,7,30,15,24,26)(3,20,35,6,32,14,21,28)(4,19,36,5,31,13,22,27)(9,10)$
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 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $QD_{16}$ x 2, $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$, 16T32, 16T48
$64$:  32T314
$144$:  $(C_3^2:C_8):C_2$
$288$:  18T110

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $D_{4}$

Degree 6: None

Degree 9: $(C_3^2:C_8):C_2$

Degree 12: None

Degree 18: 18T110

Low degree siblings

36T902 x 4, 36T939 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 $1^{36}$ $1$ $1$ $0$ $()$
2A $2^{18}$ $1$ $2$ $18$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B $2^{18}$ $2$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)$
2C $2^{18}$ $9$ $2$ $18$ $( 1,34)( 2,33)( 3,36)( 4,35)( 5, 6)( 7, 8)( 9,27)(10,28)(11,25)(12,26)(13,22)(14,21)(15,23)(16,24)(17,30)(18,29)(19,31)(20,32)$
2D $2^{16},1^{4}$ $9$ $2$ $16$ $( 1,17)( 2,18)( 3,19)( 4,20)( 5,13)( 6,14)( 7,15)( 8,16)( 9,36)(10,35)(11,34)(12,33)(25,29)(26,30)(27,32)(28,31)$
2E $2^{18}$ $18$ $2$ $18$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,23)( 6,24)( 7,22)( 8,21)( 9,18)(10,17)(11,20)(12,19)(13,16)(14,15)(29,35)(30,36)(31,34)(32,33)$
2F $2^{15},1^{6}$ $24$ $2$ $15$ $( 1,23)( 2,24)( 3,22)( 4,21)( 5,28)( 6,27)( 7,26)( 8,25)( 9,10)(13,14)(19,20)(29,33)(30,34)(31,35)(32,36)$
3A $3^{12}$ $8$ $3$ $24$ $( 1,16,25)( 2,15,26)( 3,13,27)( 4,14,28)( 5,19,32)( 6,20,31)( 7,18,30)( 8,17,29)( 9,22,36)(10,21,35)(11,24,34)(12,23,33)$
4A $4^{8},1^{4}$ $18$ $4$ $24$ $( 1,25,17,29)( 2,26,18,30)( 3,27,19,32)( 4,28,20,31)( 5,10,13,35)( 6, 9,14,36)( 7,11,15,34)( 8,12,16,33)$
4B $4^{8},2^{2}$ $18$ $4$ $26$ $( 1,34,16,18)( 2,33,15,17)( 3,36,13,20)( 4,35,14,19)( 5,31,10,22)( 6,32, 9,21)( 7,29,11,23)( 8,30,12,24)(25,26)(27,28)$
4C $4^{9}$ $24$ $4$ $27$ $( 1,31, 2,32)( 3,29, 4,30)( 5,16, 6,15)( 7,13, 8,14)( 9,11,10,12)(17,28,18,27)(19,25,20,26)(21,23,22,24)(33,36,34,35)$
4D $4^{8},1^{4}$ $36$ $4$ $24$ $( 1,25, 8,17)( 2,26, 7,18)( 3,27, 5,19)( 4,28, 6,20)( 9,31,22,14)(10,32,21,13)(11,30,24,15)(12,29,23,16)$
4E $4^{8},2^{2}$ $36$ $4$ $26$ $( 1,35,16,19)( 2,36,15,20)( 3,33,13,17)( 4,34,14,18)( 5,29,10,23)( 6,30, 9,24)( 7,31,11,22)( 8,32,12,21)(25,27)(26,28)$
4F $4^{8},2^{2}$ $36$ $4$ $26$ $( 1,26,29, 7)( 2,25,30, 8)( 3,28,32, 6)( 4,27,31, 5)( 9,10)(11,12)(13,22,19,36)(14,21,20,35)(15,23,18,33)(16,24,17,34)$
4G $4^{8},2^{2}$ $36$ $4$ $26$ $( 1,36,12,14)( 2,35,11,13)( 3,34,10,15)( 4,33, 9,16)( 5,24,19,26)( 6,23,20,25)( 7,21,18,27)( 8,22,17,28)(29,31)(30,32)$
4H $4^{8},2^{2}$ $36$ $4$ $26$ $( 1,27,29, 5)( 2,28,30, 6)( 3,25,32, 8)( 4,26,31, 7)( 9,11)(10,12)(13,23,19,33)(14,24,20,34)(15,22,18,36)(16,21,17,35)$
6A $6^{6}$ $8$ $6$ $30$ $( 1,26,16, 2,25,15)( 3,28,13, 4,27,14)( 5,31,19, 6,32,20)( 7,29,18, 8,30,17)( 9,35,22,10,36,21)(11,33,24,12,34,23)$
6B $6^{6}$ $16$ $6$ $30$ $( 1,27,16, 3,25,13)( 2,28,15, 4,26,14)( 5,29,19, 8,32,17)( 6,30,20, 7,31,18)( 9,34,22,11,36,24)(10,33,21,12,35,23)$
6C $6^{5},3^{2}$ $48$ $6$ $29$ $( 1,25,33,23, 8,29)( 2,26,34,24, 7,30)( 3,28,35,22, 5,31)( 4,27,36,21, 6,32)( 9,13,20,10,14,19)(11,15,18)(12,16,17)$
8A1 $8^{4},2,1^{2}$ $36$ $8$ $29$ $( 1,34,25, 7,17,11,29,15)( 2,33,26, 8,18,12,30,16)( 3,35,27, 5,19,10,32,13)( 4,36,28, 6,20, 9,31,14)(23,24)$
8A-1 $8^{4},2,1^{2}$ $36$ $8$ $29$ $( 1,15,29,11,17, 7,25,34)( 2,16,30,12,18, 8,26,33)( 3,13,32,10,19, 5,27,35)( 4,14,31, 9,20, 6,28,36)(23,24)$
8B1 $8^{4},4$ $36$ $8$ $31$ $( 1,32,34, 9,16,21,18, 6)( 2,31,33,10,15,22,17, 5)( 3,30,36,12,13,24,20, 8)( 4,29,35,11,14,23,19, 7)(25,27,26,28)$
8B-1 $8^{4},4$ $36$ $8$ $31$ $( 1, 6,18,21,16, 9,34,32)( 2, 5,17,22,15,10,33,31)( 3, 8,20,24,13,12,36,30)( 4, 7,19,23,14,11,35,29)(25,28,26,27)$
12A $12^{3}$ $48$ $12$ $33$ $( 1, 5,26,31,16,19, 2, 6,25,32,15,20)( 3, 7,28,29,13,18, 4, 8,27,30,14,17)( 9,23,35,11,22,33,10,24,36,12,21,34)$

Malle's constant $a(G)$:     $1/15$

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 2D 2E 2F 3A 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A1 8A-1 8B1 8B-1 12A
Size 1 1 2 9 9 18 24 8 18 18 24 36 36 36 36 36 8 16 48 36 36 36 36 48
2 P 1A 1A 1A 1A 1A 1A 1A 3A 2D 2D 2A 2D 2D 2D 2D 2D 3A 3A 3A 4A 4A 4B 4B 6A
3 P 1A 2A 2B 2C 2D 2E 2F 1A 4A 4B 4C 4D 4E 4F 4G 4H 2A 2B 2F 8A1 8A-1 8B1 8B-1 4C
Type
576.8300.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
576.8300.1b R 1 1 −1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 1 −1 −1 1 1 −1 −1 1
576.8300.1c R 1 1 −1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1 1
576.8300.1d R 1 1 −1 1 1 −1 1 1 1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 −1 1 1 −1
576.8300.1e R 1 1 −1 1 1 −1 1 1 1 1 −1 1 −1 −1 −1 1 1 −1 1 1 1 −1 −1 −1
576.8300.1f R 1 1 1 1 1 1 −1 1 1 1 −1 −1 1 −1 −1 −1 1 1 −1 1 1 1 1 −1
576.8300.1g R 1 1 1 1 1 1 −1 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
576.8300.1h R 1 1 1 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 1 1 1 −1 −1 −1 −1 1
576.8300.2a R 2 −2 0 −2 2 0 0 2 −2 2 0 −2 0 0 0 2 −2 0 0 0 0 0 0 0
576.8300.2b R 2 −2 0 −2 2 0 0 2 −2 2 0 2 0 0 0 −2 −2 0 0 0 0 0 0 0
576.8300.2c R 2 −2 0 −2 2 0 0 2 2 −2 0 0 0 −2 2 0 −2 0 0 0 0 0 0 0
576.8300.2d R 2 −2 0 −2 2 0 0 2 2 −2 0 0 0 2 −2 0 −2 0 0 0 0 0 0 0
576.8300.2e R 2 2 −2 2 2 −2 0 2 −2 −2 0 0 2 0 0 0 2 −2 0 0 0 0 0 0
576.8300.2f R 2 2 2 2 2 2 0 2 −2 −2 0 0 −2 0 0 0 2 2 0 0 0 0 0 0
576.8300.2g1 C 2 2 −2 −2 −2 2 0 2 0 0 0 0 0 0 0 0 2 −2 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83 0
576.8300.2g2 C 2 2 −2 −2 −2 2 0 2 0 0 0 0 0 0 0 0 2 −2 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83 0
576.8300.2h1 C 2 2 2 −2 −2 −2 0 2 0 0 0 0 0 0 0 0 2 2 0 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 0
576.8300.2h2 C 2 2 2 −2 −2 −2 0 2 0 0 0 0 0 0 0 0 2 2 0 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 0
576.8300.4a S 4 −4 0 4 −4 0 0 4 0 0 0 0 0 0 0 0 −4 0 0 0 0 0 0 0
576.8300.8a R 8 8 8 0 0 0 2 −1 0 0 2 0 0 0 0 0 −1 −1 −1 0 0 0 0 −1
576.8300.8b R 8 8 8 0 0 0 −2 −1 0 0 −2 0 0 0 0 0 −1 −1 1 0 0 0 0 1
576.8300.8c R 8 8 −8 0 0 0 −2 −1 0 0 2 0 0 0 0 0 −1 1 1 0 0 0 0 −1
576.8300.8d R 8 8 −8 0 0 0 2 −1 0 0 −2 0 0 0 0 0 −1 1 −1 0 0 0 0 1
576.8300.16a R 16 −16 0 0 0 0 0 −2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0

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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