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Properties

Label 36T10753
Degree $36$
Order $17496$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_3^5:F_9$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(36, 10753);
 
Copy content sage:G = TransitiveGroup(36, 10753)
 
Copy content oscar:G = transitive_group(36, 10753)
 
Copy content gap:G := TransitiveGroup(36, 10753);
 

Group invariants

Abstract group:  $C_3^5:F_9$
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 $t$:  $10753$
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:  $(1,18,27,5)(2,16,25,6)(3,17,26,4)(7,24,19,10)(8,22,20,11)(9,23,21,12)(13,29)(14,30)(15,28)(31,36)(32,34)(33,35)$, $(1,8,6,35,26,32,29,11)(2,9,4,36,27,33,30,12)(3,7,5,34,25,31,28,10)(13,19,16,23)(14,20,17,24)(15,21,18,22)$, $(1,33,18,11)(2,31,16,12)(3,32,17,10)(4,22,14,9,29,36,25,20)(5,23,15,7,30,34,26,21)(6,24,13,8,28,35,27,19)$
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)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$
$72$:  $C_3^2:C_8$ x 10
$648$:  36T1218
$1944$:  27T406

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Degree 9: None

Degree 12: 12T46

Degree 18: None

Low degree siblings

36T10753 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 $2^{12},1^{12}$ $81$ $2$ $12$ $( 1,14)( 2,15)( 3,13)( 4,28)( 5,29)( 6,30)(10,22)(11,23)(12,24)(19,33)(20,31)(21,32)$
3A $3^{12}$ $2$ $3$ $24$ $( 1, 3, 2)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,27,26)(28,30,29)(31,32,33)(34,35,36)$
3B $3^{6},1^{18}$ $4$ $3$ $12$ $( 7, 8, 9)(10,12,11)(19,20,21)(22,24,23)(31,32,33)(34,36,35)$
3C $3^{12}$ $4$ $3$ $24$ $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,20,21)(22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,36,35)$
3D1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 3, 2)(10,11,12)(13,15,14)(22,23,24)(25,27,26)(34,35,36)$
3D-1 $3^{6},1^{18}$ $4$ $3$ $12$ $( 1, 2, 3)(10,12,11)(13,14,15)(22,24,23)(25,26,27)(34,36,35)$
3E1 $3^{9},1^{9}$ $4$ $3$ $18$ $( 1, 3, 2)( 4, 6, 5)(10,12,11)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(28,30,29)(34,36,35)$
3E-1 $3^{9},1^{9}$ $4$ $3$ $18$ $( 1, 2, 3)( 4, 5, 6)(10,11,12)(13,14,15)(16,17,18)(22,23,24)(25,26,27)(28,29,30)(34,35,36)$
3F $3^{9},1^{9}$ $72$ $3$ $18$ $( 1,13,27)( 2,14,25)( 3,15,26)( 7,32,21)( 8,33,19)( 9,31,20)(10,23,34)(11,24,35)(12,22,36)$
3G $3^{6},1^{18}$ $72$ $3$ $12$ $( 4, 5, 6)(10,12,11)(19,21,20)(22,23,24)(28,30,29)(31,32,33)$
3H $3^{9},1^{9}$ $72$ $3$ $18$ $( 1,15,27)( 2,13,25)( 3,14,26)( 4,17,30)( 5,18,28)( 6,16,29)(10,35,24)(11,36,22)(12,34,23)$
3I $3^{9},1^{9}$ $72$ $3$ $18$ $( 1,13,25)( 2,14,26)( 3,15,27)( 7,31,20)( 8,32,21)( 9,33,19)(10,23,35)(11,24,36)(12,22,34)$
3J1 $3^{12}$ $72$ $3$ $24$ $( 1,14,26)( 2,15,27)( 3,13,25)( 4, 6, 5)( 7,32,21)( 8,33,19)( 9,31,20)(10,23,34)(11,24,35)(12,22,36)(16,18,17)(28,30,29)$
3J-1 $3^{12}$ $72$ $3$ $24$ $( 1,15,25)( 2,13,26)( 3,14,27)( 4, 5, 6)( 7,32,21)( 8,33,19)( 9,31,20)(10,23,34)(11,24,35)(12,22,36)(16,17,18)(28,29,30)$
3K1 $3^{9},1^{9}$ $72$ $3$ $18$ $( 1, 2, 3)(10,12,11)(13,14,15)(16,18,17)(19,21,20)(22,23,24)(25,26,27)(28,29,30)(31,32,33)$
3K-1 $3^{9},1^{9}$ $72$ $3$ $18$ $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,15,14)(19,20,21)(25,27,26)(28,30,29)(34,36,35)$
3L1 $3^{12}$ $72$ $3$ $24$ $( 1,14,25)( 2,15,26)( 3,13,27)( 4,17,30)( 5,18,28)( 6,16,29)( 7, 9, 8)(10,34,22)(11,35,23)(12,36,24)(19,21,20)(31,33,32)$
3L-1 $3^{12}$ $72$ $3$ $24$ $( 1,13,26)( 2,14,27)( 3,15,25)( 4,17,30)( 5,18,28)( 6,16,29)( 7, 8, 9)(10,36,23)(11,34,24)(12,35,22)(19,20,21)(31,32,33)$
3M1 $3^{12}$ $72$ $3$ $24$ $( 1,14,27)( 2,15,25)( 3,13,26)( 4, 6, 5)( 7,31,20)( 8,32,21)( 9,33,19)(10,23,35)(11,24,36)(12,22,34)(16,18,17)(28,30,29)$
3M-1 $3^{12}$ $72$ $3$ $24$ $( 1,15,26)( 2,13,27)( 3,14,25)( 4, 5, 6)( 7,31,20)( 8,32,21)( 9,33,19)(10,23,35)(11,24,36)(12,22,34)(16,17,18)(28,29,30)$
3N $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,25,13)( 2,26,14)( 3,27,15)( 4,30,17)( 5,28,18)( 6,29,16)( 7, 9, 8)(10,23,34)(11,24,35)(12,22,36)(19,20,21)$
3O $3^{11},1^{3}$ $216$ $3$ $22$ $( 1, 3, 2)( 4,16,30)( 5,17,28)( 6,18,29)( 7,19,33)( 8,20,31)( 9,21,32)(10,24,36)(11,22,34)(12,23,35)(13,14,15)$
3P $3^{11},1^{3}$ $216$ $3$ $22$ $( 1, 2, 3)( 4,28,16)( 5,29,17)( 6,30,18)( 7,31,20)( 8,32,21)( 9,33,19)(10,36,24)(11,34,22)(12,35,23)(25,27,26)$
3Q $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,13,26)( 2,14,27)( 3,15,25)( 4,18,28)( 5,16,29)( 6,17,30)( 7, 8, 9)(10,35,22)(11,36,23)(12,34,24)(31,33,32)$
3R $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,14,27)( 2,15,25)( 3,13,26)( 7,33,19)( 8,31,20)( 9,32,21)(10,23,34)(11,24,35)(12,22,36)(16,18,17)(28,29,30)$
3S $3^{11},1^{3}$ $216$ $3$ $22$ $( 1,14,26)( 2,15,27)( 3,13,25)( 4,29,17)( 5,30,18)( 6,28,16)( 7,19,33)( 8,20,31)( 9,21,32)(10,11,12)(34,36,35)$
4A1 $4^{6},2^{6}$ $729$ $4$ $24$ $( 1, 5,14,29)( 2, 6,15,30)( 3, 4,13,28)( 7,35)( 8,36)( 9,34)(10,32,22,21)(11,33,23,19)(12,31,24,20)(16,27)(17,25)(18,26)$
4A-1 $4^{6},2^{6}$ $729$ $4$ $24$ $( 1,29,14, 5)( 2,30,15, 6)( 3,28,13, 4)( 7,35)( 8,36)( 9,34)(10,21,22,32)(11,19,23,33)(12,20,24,31)(16,27)(17,25)(18,26)$
6A $6^{4},3^{4}$ $162$ $6$ $28$ $( 1,13, 3,15, 2,14)( 4,18, 6,17, 5,16)( 7,32, 8,33, 9,31)(10,12,11)(19,21,20)(22,35,23,36,24,34)(25,26,27)(28,29,30)$
6B $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 7,31, 8,32, 9,33)(10,24,12,23,11,22)(13,27)(14,25)(15,26)(16,29)(17,30)(18,28)(19,21,20)(34,35,36)$
6C $6^{4},3^{4}$ $324$ $6$ $28$ $( 1,27, 2,25, 3,26)( 4,16, 6,18, 5,17)( 7,20, 8,21, 9,19)(10,22,12,24,11,23)(13,15,14)(28,29,30)(31,33,32)(34,35,36)$
6D1 $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 1, 2, 3)( 4,16)( 5,17)( 6,18)(10,36,11,34,12,35)(13,25,15,27,14,26)(19,33)(20,31)(21,32)(22,24,23)$
6D-1 $6^{2},3^{2},2^{6},1^{6}$ $324$ $6$ $20$ $( 1, 3, 2)( 4,16)( 5,17)( 6,18)(10,35,12,34,11,36)(13,26,14,27,15,25)(19,33)(20,31)(21,32)(22,23,24)$
6E1 $6^{3},3^{3},2^{3},1^{3}$ $324$ $6$ $24$ $( 1, 2, 3)( 4,16, 6,18, 5,17)(10,36,12,35,11,34)(13,25,15,27,14,26)(19,32)(20,33)(21,31)(22,23,24)(28,29,30)$
6E-1 $6^{3},3^{3},2^{3},1^{3}$ $324$ $6$ $24$ $( 1, 3, 2)( 4,17, 5,18, 6,16)(10,34,11,35,12,36)(13,26,14,27,15,25)(19,32)(20,33)(21,31)(22,24,23)(28,30,29)$
8A1 $8^{3},4^{3}$ $2187$ $8$ $30$ $( 1,22, 5,21,14,10,29,32)( 2,23, 6,19,15,11,30,33)( 3,24, 4,20,13,12,28,31)( 7,26,35,18)( 8,27,36,16)( 9,25,34,17)$
8A-1 $8^{3},4^{3}$ $2187$ $8$ $30$ $( 1,32,29,10,14,21, 5,22)( 2,33,30,11,15,19, 6,23)( 3,31,28,12,13,20, 4,24)( 7,18,35,26)( 8,16,36,27)( 9,17,34,25)$
8A3 $8^{3},4^{3}$ $2187$ $8$ $30$ $( 1,21,29,22,14,32, 5,10)( 2,19,30,23,15,33, 6,11)( 3,20,28,24,13,31, 4,12)( 7,18,35,26)( 8,16,36,27)( 9,17,34,25)$
8A-3 $8^{3},4^{3}$ $2187$ $8$ $30$ $( 1,10, 5,32,14,22,29,21)( 2,11, 6,33,15,23,30,19)( 3,12, 4,31,13,24,28,20)( 7,26,35,18)( 8,27,36,16)( 9,25,34,17)$
12A1 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1, 5,13,16, 3, 4,15,18, 2, 6,14,17)( 7,24,32,34, 8,22,33,35, 9,23,31,36)(10,21,12,20,11,19)(25,29,26,30,27,28)$
12A-1 $12^{2},6^{2}$ $1458$ $12$ $32$ $( 1,17,14, 6, 2,18,15, 4, 3,16,13, 5)( 7,36,31,23, 9,35,33,22, 8,34,32,24)(10,19,11,20,12,21)(25,28,27,30,26,29)$

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

42 x 42 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