Properties

Label 33T7
33T7 1 11 1->11 19 1->19 2 12 2->12 21 2->21 3 10 3->10 20 3->20 4 14 4->14 30 4->30 5 15 5->15 29 5->29 6 13 6->13 28 6->28 7 7->4 16 7->16 8 8->6 17 8->17 9 9->5 18 9->18 10->13 10->21 11->15 11->19 12->14 12->20 23 13->23 24 13->24 22 14->22 14->22 15->23 15->24 25 16->25 31 16->31 26 17->26 33 17->33 27 18->27 32 18->32 19->8 19->28 20->7 20->29 21->9 21->30 22->16 22->33 23->18 23->31 24->17 24->32 25->3 25->26 26->1 27->2 28->2 28->4 29->1 29->5 30->3 30->6 31->7 31->10 32->8 32->12 33->9 33->11
Degree $33$
Order $330$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group no
Group: $C_{33}:C_{10}$

Related objects

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(33, 7);
 
Copy content sage:G = TransitiveGroup(33, 7)
 
Copy content oscar:G = transitive_group(33, 7)
 
Copy content gap:G := TransitiveGroup(33, 7);
 

Group invariants

Abstract group:  $C_{33}:C_{10}$
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:  $330=2 \cdot 3 \cdot 5 \cdot 11$
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 $n$:  $33$
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 $t$:  $7$
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(33).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(33), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(33), G));
 
Generators:  $(1,19,8,6,28,2,21,9,5,29)(3,20,7,4,30)(10,13,23,18,32,12,14,22,16,31)(11,15,24,17,33)(25,26)$, $(1,11,19,28,4,14,22,33,9,18,27,2,12,20,29,5,15,23,31,7,16,25,3,10,21,30,6,13,24,32,8,17,26)$
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)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$
$55$:  $C_{11}:C_5$
$110$:  22T5

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 11: $C_{11}:C_5$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{33}$ $1$ $1$ $0$ $()$
2A $2^{11},1^{11}$ $3$ $2$ $11$ $( 2, 3)( 4, 6)( 7, 9)(10,11)(14,15)(16,17)(19,20)(23,24)(26,27)(29,30)(32,33)$
3A $3^{11}$ $2$ $3$ $22$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)$
5A1 $5^{6},1^{3}$ $11$ $5$ $24$ $( 1,12, 5,21,31)( 2,10, 6,19,32)( 3,11, 4,20,33)( 7,30,27,17,24)( 8,28,25,18,22)( 9,29,26,16,23)$
5A-1 $5^{6},1^{3}$ $11$ $5$ $24$ $( 1,31,21, 5,12)( 2,32,19, 6,10)( 3,33,20, 4,11)( 7,24,17,27,30)( 8,22,18,25,28)( 9,23,16,26,29)$
5A2 $5^{6},1^{3}$ $11$ $5$ $24$ $( 1, 5,31,12,21)( 2, 6,32,10,19)( 3, 4,33,11,20)( 7,27,24,30,17)( 8,25,22,28,18)( 9,26,23,29,16)$
5A-2 $5^{6},1^{3}$ $11$ $5$ $24$ $( 1,21,12,31, 5)( 2,19,10,32, 6)( 3,20,11,33, 4)( 7,17,30,24,27)( 8,18,28,22,25)( 9,16,29,23,26)$
10A1 $10^{2},5^{2},2,1$ $33$ $10$ $27$ $( 1,22,13,31,28)( 2,24,14,33,29, 3,23,15,32,30)( 4,16,27, 9,11, 6,17,26, 7,10)( 5,18,25, 8,12)(19,20)$
10A-1 $10^{2},5^{2},2,1$ $33$ $10$ $27$ $( 1,28,31,13,22)( 2,30,32,15,23, 3,29,33,14,24)( 4,10, 7,26,17, 6,11, 9,27,16)( 5,12, 8,25,18)(19,20)$
10A3 $10^{2},5^{2},2,1$ $33$ $10$ $27$ $( 1,31,22,28,13)( 2,33,23,30,14, 3,32,24,29,15)( 4, 9,17,10,27, 6, 7,16,11,26)( 5, 8,18,12,25)(19,20)$
10A-3 $10^{2},5^{2},2,1$ $33$ $10$ $27$ $( 1,13,28,22,31)( 2,15,29,24,32, 3,14,30,23,33)( 4,26,11,16, 7, 6,27,10,17, 9)( 5,25,12,18, 8)(19,20)$
11A1 $11^{3}$ $5$ $11$ $30$ $( 1,18,31,13,28,12,25, 8,22, 5,21)( 2,16,32,14,29,10,26, 9,23, 6,19)( 3,17,33,15,30,11,27, 7,24, 4,20)$
11A-1 $11^{3}$ $5$ $11$ $30$ $( 1,21, 5,22, 8,25,12,28,13,31,18)( 2,19, 6,23, 9,26,10,29,14,32,16)( 3,20, 4,24, 7,27,11,30,15,33,17)$
15A1 $15^{2},3$ $22$ $15$ $30$ $( 1, 4,32,12,20, 2, 5,33,10,21, 3, 6,31,11,19)( 7,26,22,30,16, 8,27,23,28,17, 9,25,24,29,18)(13,15,14)$
15A-1 $15^{2},3$ $22$ $15$ $30$ $( 1,19,11,31, 6, 3,21,10,33, 5, 2,20,12,32, 4)( 7,18,29,24,25, 9,17,28,23,27, 8,16,30,22,26)(13,14,15)$
15A2 $15^{2},3$ $22$ $15$ $30$ $( 1,32,20, 5,10, 3,31,19, 4,12, 2,33,21, 6,11)( 7,22,16,27,28, 9,24,18,26,30, 8,23,17,25,29)(13,14,15)$
15A-2 $15^{2},3$ $22$ $15$ $30$ $( 1,11, 6,21,33, 2,12, 4,19,31, 3,10, 5,20,32)( 7,29,25,17,23, 8,30,26,18,24, 9,28,27,16,22)(13,15,14)$
22A1 $22,11$ $15$ $22$ $31$ $( 1,25,18, 8,31,22,13, 5,28,21,12)( 2,27,16, 7,32,24,14, 4,29,20,10, 3,26,17, 9,33,23,15, 6,30,19,11)$
22A-1 $22,11$ $15$ $22$ $31$ $( 1,18,31,13,28,12,25, 8,22, 5,21)( 2,17,32,15,29,11,26, 7,23, 4,19, 3,16,33,14,30,10,27, 9,24, 6,20)$
33A1 $33$ $10$ $33$ $32$ $( 1,33,29,25,24,19,18,15,10, 8, 4, 2,31,30,26,22,20,16,13,11, 9, 5, 3,32,28,27,23,21,17,14,12, 7, 6)$
33A-1 $33$ $10$ $33$ $32$ $( 1,29,24,18,10, 4,31,26,20,13, 9, 3,28,23,17,12, 6,33,25,19,15, 8, 2,30,22,16,11, 5,32,27,21,14, 7)$

Malle's constant $a(G)$:     $1/11$

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 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 11A1 11A-1 15A1 15A-1 15A2 15A-2 22A1 22A-1 33A1 33A-1
Size 1 3 2 11 11 11 11 33 33 33 33 5 5 22 22 22 22 15 15 10 10
2 P 1A 1A 3A 5A2 5A-2 5A-1 5A1 5A-1 5A1 5A2 5A-2 11A-1 11A1 15A2 15A-2 15A-1 15A1 11A1 11A-1 33A-1 33A1
3 P 1A 2A 1A 5A-2 5A2 5A1 5A-1 10A3 10A-3 10A-1 10A1 11A1 11A-1 5A1 5A-1 5A2 5A-2 22A1 22A-1 11A-1 11A1
5 P 1A 2A 3A 1A 1A 1A 1A 2A 2A 2A 2A 11A1 11A-1 3A 3A 3A 3A 22A1 22A-1 33A1 33A-1
11 P 1A 2A 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 1A 1A 15A1 15A-1 15A2 15A-2 2A 2A 3A 3A
Type
330.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
330.2.1b R 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 1 1
330.2.1c1 C 1 1 1 ζ5−2 ζ5 ζ52 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 1 1 1
330.2.1c2 C 1 1 1 ζ52 ζ5−1 ζ5−2 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 1 1 1
330.2.1c3 C 1 1 1 ζ5−1 ζ5−2 ζ5 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 1 1 1
330.2.1c4 C 1 1 1 ζ5 ζ52 ζ5−1 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 1 1 1
330.2.1d1 C 1 −1 1 ζ5−2 ζ5 ζ52 ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 −1 −1 1 1
330.2.1d2 C 1 −1 1 ζ52 ζ5−1 ζ5−2 ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 1 ζ5−2 ζ52 ζ5 ζ5−1 −1 −1 1 1
330.2.1d3 C 1 −1 1 ζ5−1 ζ5−2 ζ5 ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 1 ζ5 ζ5−1 ζ52 ζ5−2 −1 −1 1 1
330.2.1d4 C 1 −1 1 ζ5 ζ52 ζ5−1 ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 1 ζ5−1 ζ5 ζ5−2 ζ52 −1 −1 1 1
330.2.2a R 2 0 −1 2 2 2 2 0 0 0 0 2 2 −1 −1 −1 −1 0 0 −1 −1
330.2.2b1 C 2 0 −1 2ζ5−2 2ζ5 2ζ52 2ζ5−1 0 0 0 0 2 2 −ζ52 −ζ5−2 −ζ5−1 −ζ5 0 0 −1 −1
330.2.2b2 C 2 0 −1 2ζ52 2ζ5−1 2ζ5−2 2ζ5 0 0 0 0 2 2 −ζ5−2 −ζ52 −ζ5 −ζ5−1 0 0 −1 −1
330.2.2b3 C 2 0 −1 2ζ5−1 2ζ5−2 2ζ5 2ζ52 0 0 0 0 2 2 −ζ5 −ζ5−1 −ζ52 −ζ5−2 0 0 −1 −1
330.2.2b4 C 2 0 −1 2ζ5 2ζ52 2ζ5−1 2ζ5−2 0 0 0 0 2 2 −ζ5−1 −ζ5 −ζ5−2 −ζ52 0 0 −1 −1
330.2.5a1 C 5 5 5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
330.2.5a2 C 5 5 5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
330.2.5b1 C 5 −5 5 0 0 0 0 0 0 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 0 0 0 0 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
330.2.5b2 C 5 −5 5 0 0 0 0 0 0 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 0 0 0 0 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
330.2.10a1 C 10 0 −5 0 0 0 0 0 0 0 0 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 0 0 0 0 0 0 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115
330.2.10a2 C 10 0 −5 0 0 0 0 0 0 0 0 2ζ11−2+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−2−2ζ11−2ζ113−2ζ114−2ζ115 0 0 0 0 0 0 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115

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