Properties

Label 36T902
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, 902);
 
Copy content sage:G = TransitiveGroup(36, 902)
 
Copy content oscar:G = transitive_group(36, 902)
 
Copy content gap:G := TransitiveGroup(36, 902);
 

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 $t$:  $902$
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:  $(1,13,17,28,10,35,7,22)(2,14,18,27,9,36,8,21)(3,16,19,26,11,33,5,23)(4,15,20,25,12,34,6,24)(29,31,30,32)$, $(1,27,18,5,24,12)(2,28,17,6,23,11)(3,26,20,7,22,9)(4,25,19,8,21,10)(13,33,29,16,35,32)(14,34,30,15,36,31)$, $(1,20,2,19)(3,17,4,18)(5,10,6,9)(7,12,8,11)(13,33,14,34)(15,35,16,36)(21,24,22,23)(25,28,26,27)(29,32,30,31)$
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: 18T68

Low degree siblings

36T902 x 3, 36T939 x 4

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

Malle's constant $a(G)$:     $1/9$

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