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Properties

Label 36T10766
Degree $36$
Order $17496$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^5:\PSU(3,2)$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 10766);
 
Copy content sage:G = TransitiveGroup(36, 10766)
 
Copy content oscar:G = transitive_group(36, 10766)
 
Copy content gap:G := TransitiveGroup(36, 10766);
 

Group invariants

Abstract group:  $C_3^5:\PSU(3,2)$
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:  $17496=2^{3} \cdot 3^{7}$
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 $n$:  $36$
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 $t$:  $10766$
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);
 
$\card{\Aut(F/K)}$:  $3$
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:  $(1,31,26,9)(2,33,27,8)(3,32,25,7)(4,36,30,10)(5,34,28,11)(6,35,29,12)(13,20)(14,19)(15,21)(16,22)(17,23)(18,24)$, $(1,10,3,11,2,12)(4,21,30,8,5,19,28,9,6,20,29,7)(13,35,27,22,14,34,25,24,15,36,26,23)(16,31,18,33,17,32)$
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
$4$:  $C_2^2$
$8$:  $Q_8$
$72$:  $C_3^2:Q_8$ x 4
$648$:  12T174
$1944$:  27T420 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Degree 9: None

Degree 12: 12T47

Degree 18: None

Low degree siblings

36T10766 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 $2^{12},1^{12}$ $81$ $2$ $12$ $( 4,18)( 5,16)( 6,17)(10,34)(11,35)(12,36)(13,26)(14,27)(15,25)(19,32)(20,33)(21,31)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,21,20)(22,23,24)(25,27,26)(28,30,29)(31,33,32)(34,35,36)$
3B $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,24,23)(25,27,26)(28,29,30)(31,33,32)(34,36,35)$
3C $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26)(28,30,29)(31,32,33)(34,36,35)$
3D $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 2, 3)(10,11,12)(13,14,15)(22,23,24)(25,26,27)(34,35,36)$
3E $3^{6},1^{18}$ $4$ $3$ $12$ $( 4, 6, 5)(10,11,12)(16,18,17)(22,23,24)(28,30,29)(34,35,36)$
3F $3^{6},1^{18}$ $4$ $3$ $12$ $( 7, 9, 8)(10,11,12)(19,21,20)(22,23,24)(31,33,32)(34,35,36)$
3G1 $3^{9},1^{9}$ $4$ $3$ $18$ $( 1, 2, 3)( 7, 9, 8)(10,12,11)(13,14,15)(19,21,20)(22,24,23)(25,26,27)(31,33,32)(34,36,35)$
3G-1 $3^{9},1^{9}$ $4$ $3$ $18$ $( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,15,14)(19,20,21)(22,23,24)(25,27,26)(31,32,33)(34,35,36)$
3H $3^{6},1^{18}$ $72$ $3$ $12$ $(16,17,18)(19,20,21)(22,23,24)(28,30,29)(31,33,32)(34,36,35)$
3I $3^{9},1^{9}$ $72$ $3$ $18$ $( 4,16,28)( 5,17,29)( 6,18,30)( 7,19,32)( 8,20,33)( 9,21,31)(10,24,35)(11,22,36)(12,23,34)$
3J $3^{9},1^{9}$ $72$ $3$ $18$ $( 4,16,29)( 5,17,30)( 6,18,28)( 7,19,33)( 8,20,31)( 9,21,32)(10,24,36)(11,22,34)(12,23,35)$
3K $3^{9},1^{9}$ $72$ $3$ $18$ $( 4,16,30)( 5,17,28)( 6,18,29)( 7,19,31)( 8,20,32)( 9,21,33)(10,24,34)(11,22,35)(12,23,36)$
3L1 $3^{9},1^{9}$ $72$ $3$ $18$ $(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,35,36)$
3L-1 $3^{9},1^{9}$ $72$ $3$ $18$ $(10,12,11)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,26,27)(28,29,30)(31,33,32)(34,36,35)$
3M1 $3^{12}$ $72$ $3$ $24$ $( 1, 2, 3)( 4,16,28)( 5,17,29)( 6,18,30)( 7,19,32)( 8,20,33)( 9,21,31)(10,22,34)(11,23,35)(12,24,36)(13,14,15)(25,26,27)$
3M-1 $3^{12}$ $72$ $3$ $24$ $( 1, 3, 2)( 4,16,28)( 5,17,29)( 6,18,30)( 7,19,32)( 8,20,33)( 9,21,31)(10,23,36)(11,24,34)(12,22,35)(13,15,14)(25,27,26)$
3N1 $3^{12}$ $72$ $3$ $24$ $( 1, 2, 3)( 4,16,29)( 5,17,30)( 6,18,28)( 7,19,33)( 8,20,31)( 9,21,32)(10,22,35)(11,23,36)(12,24,34)(13,14,15)(25,26,27)$
3N-1 $3^{12}$ $72$ $3$ $24$ $( 1, 3, 2)( 4,16,29)( 5,17,30)( 6,18,28)( 7,19,33)( 8,20,31)( 9,21,32)(10,23,34)(11,24,35)(12,22,36)(13,15,14)(25,27,26)$
3O1 $3^{12}$ $72$ $3$ $24$ $( 1, 2, 3)( 4,16,30)( 5,17,28)( 6,18,29)( 7,19,31)( 8,20,32)( 9,21,33)(10,22,36)(11,23,34)(12,24,35)(13,14,15)(25,26,27)$
3O-1 $3^{12}$ $72$ $3$ $24$ $( 1, 3, 2)( 4,16,30)( 5,17,28)( 6,18,29)( 7,19,31)( 8,20,32)( 9,21,33)(10,23,35)(11,24,36)(12,22,34)(13,15,14)(25,27,26)$
3P $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,25,15)( 2,26,13)( 3,27,14)( 4, 6, 5)( 7,19,31)( 8,20,32)( 9,21,33)(10,34,22)(11,35,23)(12,36,24)(28,29,30)$
3Q $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,14,25)( 2,15,26)( 3,13,27)( 4,18,29)( 5,16,30)( 6,17,28)( 7, 9, 8)(10,36,24)(11,34,22)(12,35,23)(31,32,33)$
3R $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,25,13)( 2,26,14)( 3,27,15)( 4, 6, 5)( 7,19,33)( 8,20,31)( 9,21,32)(10,36,22)(11,34,23)(12,35,24)(28,29,30)$
3S $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,25,14)( 2,26,15)( 3,27,13)( 4, 5, 6)( 7,20,31)( 8,21,32)( 9,19,33)(10,35,22)(11,36,23)(12,34,24)(16,18,17)$
3T $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,25,15)( 2,26,13)( 3,27,14)( 4,18,29)( 5,16,30)( 6,17,28)( 7,33,20)( 8,31,21)( 9,32,19)(10,11,12)(34,36,35)$
3U $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,27,14)( 2,25,15)( 3,26,13)( 7,21,32)( 8,19,33)( 9,20,31)(10,35,22)(11,36,23)(12,34,24)(16,18,17)(28,29,30)$
4A $4^{6},2^{6}$ $1458$ $4$ $24$ $( 1,23)( 2,22)( 3,24)( 4,20,18,33)( 5,21,16,31)( 6,19,17,32)( 7,28)( 8,29)( 9,30)(10,25,34,15)(11,27,35,14)(12,26,36,13)$
4B $4^{6},2^{6}$ $1458$ $4$ $24$ $( 1,16)( 2,18)( 3,17)( 4,27,28,14)( 5,26,29,13)( 6,25,30,15)( 7,23,20,36)( 8,24,21,34)( 9,22,19,35)(10,31)(11,32)(12,33)$
4C $4^{6},2^{6}$ $1458$ $4$ $24$ $( 1,19,13,33)( 2,21,14,32)( 3,20,15,31)( 4,22,30,10)( 5,23,28,11)( 6,24,29,12)( 7,27)( 8,26)( 9,25)(16,36)(17,34)(18,35)$
6A $6^{4},3^{4}$ $162$ $6$ $28$ $( 1, 2, 3)( 4,16, 6,18, 5,17)( 7, 8, 9)(10,36,11,34,12,35)(13,27,15,26,14,25)(19,33,21,32,20,31)(22,24,23)(28,29,30)$
6B $6^{4},3^{4}$ $162$ $6$ $28$ $( 1, 2, 3)( 4,30, 5,28, 6,29)( 7,21, 9,20, 8,19)(10,11,12)(13,27,15,26,14,25)(16,18,17)(22,36,24,35,23,34)(31,32,33)$
6C $6^{4},3^{4}$ $162$ $6$ $28$ $( 1,14, 3,13, 2,15)( 4,28, 6,30, 5,29)( 7, 9, 8)(10,23,12,22,11,24)(16,17,18)(19,32,20,33,21,31)(25,26,27)(34,35,36)$
6D $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 1, 3, 2)( 7,32)( 8,33)( 9,31)(10,24,11,22,12,23)(13,25,14,26,15,27)(16,30)(17,28)(18,29)(34,36,35)$
6E $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 1,13)( 2,14)( 3,15)( 4,30, 6,29, 5,28)(10,22,11,23,12,24)(16,17,18)(19,31)(20,32)(21,33)(34,36,35)$
6F $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 1,26)( 2,27)( 3,25)( 4,16)( 5,17)( 6,18)( 7,19, 9,21, 8,20)(10,24,11,22,12,23)(31,32,33)(34,36,35)$
6G1 $6^{3},3^{3},2^{3},1^{3}$ $324$ $6$ $24$ $( 1, 3, 2)( 7,33, 9,32, 8,31)(10,24,12,23,11,22)(13,26,14,27,15,25)(16,29)(17,30)(18,28)(19,20,21)(34,35,36)$
6G-1 $6^{3},3^{3},2^{3},1^{3}$ $324$ $6$ $24$ $( 1, 2, 3)( 7,31, 8,32, 9,33)(10,22,11,23,12,24)(13,25,15,27,14,26)(16,29)(17,30)(18,28)(19,21,20)(34,36,35)$
12A1 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,24, 2,23, 3,22)( 4,32,16,20, 6,31,18,19, 5,33,17,21)( 7,30, 8,28, 9,29)(10,14,36,25,11,13,34,27,12,15,35,26)$
12A5 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,22, 3,23, 2,24)( 4,31,17,20, 5,32,18,21, 6,33,16,19)( 7,29, 9,28, 8,30)(10,13,35,25,12,14,34,26,11,15,36,27)$
12B1 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,17, 2,16, 3,18)( 4,13,30,27, 5,15,28,26, 6,14,29,25)( 7,35,21,23, 9,34,20,22, 8,36,19,24)(10,33,11,31,12,32)$
12B5 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,18, 3,16, 2,17)( 4,15,29,27, 6,13,28,25, 5,14,30,26)( 7,34,19,23, 8,35,20,24, 9,36,21,22)(10,32,12,31,11,33)$
12C1 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,31,14,19, 3,32,13,20, 2,33,15,21)( 4,12,28,22, 6,11,30,24, 5,10,29,23)( 7,26, 9,27, 8,25)(16,35,17,36,18,34)$
12C5 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,32,15,19, 2,31,13,21, 3,33,14,20)( 4,11,29,22, 5,12,30,23, 6,10,28,24)( 7,25, 8,27, 9,26)(16,34,18,36,17,35)$

Malle's constant $a(G)$:     $1/12$

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

45 x 45 character table

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