Properties

Label 36T2617
Degree $36$
Order $1728$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_6^2:\GL(2,3)$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 2617);
 
Copy content sage:G = TransitiveGroup(36, 2617)
 
Copy content oscar:G = transitive_group(36, 2617)
 
Copy content gap:G := TransitiveGroup(36, 2617);
 

Group invariants

Abstract group:  $C_6^2:\GL(2,3)$
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:  $1728=2^{6} \cdot 3^{3}$
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 $t$:  $2617$
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);
 
$\card{\Aut(F/K)}$:  $1$
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:  $(1,16,10)(2,14,9)(3,13,12)(4,15,11)(5,24,33)(6,22,35)(7,21,36)(8,23,34)(17,31,26)(18,30,28)(19,32,27)(20,29,25)$, $(1,6,35,4,5,36,2,7,33,3,8,34)(9,28,17,21,13,30,11,26,20,24,14,29)(10,25,19,22,16,32,12,27,18,23,15,31)$
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)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$ x 3
$48$:  $\textrm{GL(2,3)}$
$96$:  $V_4^2:S_3$
$192$:  24T314
$432$:  $((C_3^2:Q_8):C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: $S_4$

Degree 6: None

Degree 9: $((C_3^2:Q_8):C_3):C_2$

Degree 12: None

Degree 18: None

Low degree siblings

36T2617

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 $2^{18}$ $3$ $2$ $18$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)$
2B $2^{16},1^{4}$ $9$ $2$ $16$ $( 1,28)( 2,26)( 3,25)( 4,27)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,18)(11,17)(12,19)(29,35)(30,33)(31,34)(32,36)$
2C $2^{18}$ $27$ $2$ $18$ $( 1, 3)( 2, 4)( 5,36)( 6,33)( 7,35)( 8,34)( 9,29)(10,32)(11,30)(12,31)(13,26)(14,28)(15,27)(16,25)(17,24)(18,23)(19,22)(20,21)$
2D $2^{15},1^{6}$ $72$ $2$ $15$ $( 1,10)( 2,12)( 3, 9)( 4,11)( 5,24)( 6,22)( 7,23)( 8,21)(13,14)(17,27)(18,28)(19,26)(20,25)(31,32)(34,36)$
3A $3^{12}$ $8$ $3$ $24$ $( 1,24,18)( 2,21,19)( 3,23,17)( 4,22,20)( 5,28,10)( 6,25,11)( 7,27, 9)( 8,26,12)(13,34,31)(14,36,32)(15,35,29)(16,33,30)$
3B $3^{11},1^{3}$ $96$ $3$ $22$ $( 1,21,17)( 2,23,18)( 3,24,19)( 4,22,20)( 5,12,25)( 6,10,26)( 7, 9,27)( 8,11,28)(14,16,15)(29,32,30)(33,35,36)$
3C $3^{12}$ $192$ $3$ $24$ $( 1,17, 7)( 2,19, 8)( 3,20, 5)( 4,18, 6)( 9,28,14)(10,25,13)(11,27,16)(12,26,15)(21,29,35)(22,30,36)(23,31,33)(24,32,34)$
4A $4^{8},1^{4}$ $54$ $4$ $24$ $( 1,24,28, 5)( 2,21,26, 8)( 3,23,25, 6)( 4,22,27, 7)( 9,34,20,31)(10,33,18,30)(11,36,17,32)(12,35,19,29)$
4B $4^{8},2^{2}$ $54$ $4$ $26$ $( 1,26,24,35)( 2,28,21,33)( 3,27,23,34)( 4,25,22,36)( 5,15,30,12)( 6,13,32, 9)( 7,14,31,11)( 8,16,29,10)(17,20)(18,19)$
4C $4^{9}$ $72$ $4$ $27$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,13, 6,15)( 7,14, 8,16)(17,35,18,34)(19,33,20,36)(21,24,22,23)(25,26,28,27)(29,30,31,32)$
4D $4^{8},2^{2}$ $108$ $4$ $26$ $( 1,14,18, 6)( 2,13,19, 7)( 3,16,17, 5)( 4,15,20, 8)( 9,29,34,26)(10,32,33,25)(11,30,36,28)(12,31,35,27)(21,22)(23,24)$
6A $6^{6}$ $24$ $6$ $30$ $( 1,36, 5, 3,33, 6)( 2,34, 8, 4,35, 7)( 9,19,13,12,20,15)(10,17,16,11,18,14)(21,31,26,22,29,27)(23,30,25,24,32,28)$
6B $6^{5},3^{2}$ $144$ $6$ $29$ $( 1,28,24,10,18, 5)( 2,26,21,12,19, 8)( 3,27,23, 9,17, 7)( 4,25,22,11,20, 6)(13,32,34,14,31,36)(15,29,35)(16,30,33)$
6C $6^{5},3,2,1$ $288$ $6$ $28$ $( 1,11,21,28,17, 8)( 2,10,23,26,18, 6)( 3,12,24,25,19, 5)( 4, 9,22,27,20, 7)(14,15,16)(29,33,32,35,30,36)(31,34)$
8A1 $8^{4},2,1^{2}$ $108$ $8$ $29$ $( 1,31,24, 9,28,34, 5,20)( 2,29,21,12,26,35, 8,19)( 3,32,23,11,25,36, 6,17)( 4,30,22,10,27,33, 7,18)(13,16)$
8A-1 $8^{4},2,1^{2}$ $108$ $8$ $29$ $( 1,20, 5,34,28, 9,24,31)( 2,19, 8,35,26,12,21,29)( 3,17, 6,36,25,11,23,32)( 4,18, 7,33,27,10,22,30)(13,16)$
8B1 $8^{4},4$ $108$ $8$ $31$ $( 1,32,26, 9,24, 6,35,13)( 2,31,28,11,21, 7,33,14)( 3,29,27,10,23, 8,34,16)( 4,30,25,12,22, 5,36,15)(17,19,20,18)$
8B-1 $8^{4},4$ $108$ $8$ $31$ $( 1,13,35, 6,24, 9,26,32)( 2,14,33, 7,21,11,28,31)( 3,16,34, 8,23,10,27,29)( 4,15,36, 5,22,12,25,30)(17,18,20,19)$
12A $12^{3}$ $144$ $12$ $33$ $( 1,15,36, 9, 5,19, 3,13,33,12, 6,20)( 2,14,34,10, 8,17, 4,16,35,11, 7,18)(21,25,31,24,26,32,22,28,29,23,27,30)$

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 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 8A1 8A-1 8B1 8B-1 12A
Size 1 3 9 27 72 8 96 192 54 54 72 108 24 144 288 108 108 108 108 144
2 P 1A 1A 1A 1A 1A 3A 3B 3C 2B 2B 2A 2B 3A 3A 3B 4A 4A 4B 4B 6A
3 P 1A 2A 2B 2C 2D 1A 1A 1A 4A 4B 4C 4D 2A 2D 2B 8A1 8A-1 8B1 8B-1 4C
Type
1728.47846.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1728.47846.1b R 1 1 1 1 −1 1 1 1 1 1 −1 1 1 −1 1 −1 −1 −1 −1 −1
1728.47846.2a R 2 2 2 2 0 2 −1 −1 2 2 0 2 2 0 −1 0 0 0 0 0
1728.47846.2b1 C 2 2 −2 −2 0 2 −1 −1 0 0 0 0 2 0 1 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 0
1728.47846.2b2 C 2 2 −2 −2 0 2 −1 −1 0 0 0 0 2 0 1 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 0
1728.47846.3a R 3 −1 3 −1 1 3 0 0 −1 3 −1 −1 −1 1 0 −1 −1 1 1 −1
1728.47846.3b R 3 −1 3 −1 1 3 0 0 3 −1 −1 −1 −1 1 0 1 1 −1 −1 −1
1728.47846.3c R 3 3 3 3 1 3 0 0 −1 −1 1 −1 3 1 0 −1 −1 −1 −1 1
1728.47846.3d R 3 −1 3 −1 −1 3 0 0 −1 3 1 −1 −1 −1 0 1 1 −1 −1 1
1728.47846.3e R 3 −1 3 −1 −1 3 0 0 3 −1 1 −1 −1 −1 0 −1 −1 1 1 1
1728.47846.3f R 3 3 3 3 −1 3 0 0 −1 −1 −1 −1 3 −1 0 1 1 1 1 −1
1728.47846.4a R 4 4 −4 −4 0 4 1 1 0 0 0 0 4 0 −1 0 0 0 0 0
1728.47846.6a R 6 −2 6 −2 0 6 0 0 −2 −2 0 2 −2 0 0 0 0 0 0 0
1728.47846.6b1 C 6 −2 −6 2 0 6 0 0 0 0 0 0 −2 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83 0
1728.47846.6b2 C 6 −2 −6 2 0 6 0 0 0 0 0 0 −2 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83 0
1728.47846.8a R 8 8 0 0 2 −1 2 −1 0 0 2 0 −1 −1 0 0 0 0 0 −1
1728.47846.8b R 8 8 0 0 −2 −1 2 −1 0 0 −2 0 −1 1 0 0 0 0 0 1
1728.47846.16a R 16 16 0 0 0 −2 −2 1 0 0 0 0 −2 0 0 0 0 0 0 0
1728.47846.24a R 24 −8 0 0 2 −3 0 0 0 0 −2 0 1 −1 0 0 0 0 0 1
1728.47846.24b R 24 −8 0 0 −2 −3 0 0 0 0 2 0 1 1 0 0 0 0 0 −1

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