Properties

Label 38T33
38T33 1 21 1->21 2 2->1 3 7 3->7 20 3->20 4 10 4->10 5 5->4 13 5->13 6 30 6->30 31 6->31 17 7->17 7->20 8 9 8->9 29 8->29 9->13 36 9->36 26 10->26 35 10->35 11 11->5 11->9 12 14 12->14 33 12->33 13->11 22 13->22 16 14->16 14->17 15 15->6 15->35 24 16->24 16->29 17->12 25 17->25 18 18->10 18->21 19 19->22 28 19->28 20->3 23 20->23 21->2 38 21->38 22->14 22->24 23->11 23->31 24->19 34 24->34 25->23 27 25->27 26->8 26->18 27->19 27->28 32 28->32 28->38 29->7 29->33 30->18 37 30->37 31->25 31->37 32->30 33->15 33->16 34->12 34->15 35->26 35->34 36->3 36->8 37->6 37->36 38->5 38->27
Degree $38$
Order $25308$
Cyclic no
Abelian no
Solvable no
Transitivity $2$
Primitive yes
$p$-group no
Group: $\PSL(2,37)$

Related objects

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(38, 33);
 
Copy content sage:G = TransitiveGroup(38, 33)
 
Copy content oscar:G = transitive_group(38, 33)
 
Copy content gap:G := TransitiveGroup(38, 33);
 

Group invariants

Abstract group:  $\PSL(2,37)$
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:  $25308=2^{2} \cdot 3^{2} \cdot 19 \cdot 37$
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:  no
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 $n$:  $38$
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 $t$:  $33$
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:  2
Primitive:  yes
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(38).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(38), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(38), G));
 
Generators:  $(3,20,23,11,9,13,22,14,16,24,34,12,33,15,6,31,37,36)(4,10,35,26,8,29,7,17,25,27,19,28,32,30,18,21,38,5)$, $(1,21,2)(3,7,20)(5,13,11)(6,30,37)(8,9,36)(10,26,18)(12,14,17)(15,35,34)(16,29,33)(19,22,24)(23,31,25)(27,28,38)$
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

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 19: None

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 $1^{38}$ $1$ $1$ $0$ $()$
2A $2^{18},1^{2}$ $703$ $2$ $18$ $( 1,37)( 2,30)( 3,16)( 4,36)( 5,38)( 7,34)( 8, 9)(10,13)(11,24)(12,32)(14,17)(15,33)(18,19)(20,31)(22,27)(23,29)(25,35)(26,28)$
3A $3^{12},1^{2}$ $1406$ $3$ $24$ $( 1,25,16)( 2,26,11)( 3,37,35)( 4,15,27)( 5,32,29)( 7,14,19)( 8,13,20)( 9,10,31)(12,23,38)(17,18,34)(22,36,33)(24,30,28)$
6A $6^{6},1^{2}$ $1406$ $6$ $30$ $( 1, 3,25,37,16,35)( 2,24,26,30,11,28)( 4,22,15,36,27,33)( 5,23,32,38,29,12)( 7,18,14,34,19,17)( 8,31,13, 9,20,10)$
9A1 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,13,36,25,20,33,16, 8,22)( 2,29,14,26, 5,19,11,32, 7)( 3, 9,27,37,10, 4,35,31,15)(12,34,30,23,17,28,38,18,24)$
9A2 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,36,20,16,22,13,25,33, 8)( 2,14, 5,11, 7,29,26,19,32)( 3,27,10,35,15, 9,37, 4,31)(12,30,17,38,24,34,23,28,18)$
9A4 $9^{4},1^{2}$ $1406$ $9$ $32$ $( 1,20,22,25, 8,36,16,13,33)( 2, 5, 7,26,32,14,11,29,19)( 3,10,15,37,31,27,35, 9, 4)(12,17,24,23,18,30,38,34,28)$
18A1 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1,15,13, 3,36, 9,25,27,20,37,33,10,16, 4, 8,35,22,31)( 2,18,29,24,14,12,26,34, 5,30,19,23,11,17,32,28, 7,38)$
18A5 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1, 9,33,35,13,27,16,31,36,37, 8,15,25,10,22, 3,20, 4)( 2,12,19,28,29,34,11,38,14,30,32,18,26,23, 7,24, 5,17)$
18A7 $18^{2},1^{2}$ $1406$ $18$ $34$ $( 1,27, 8, 3,33,31,25, 4,13,37,22, 9,16,15,20,35,36,10)( 2,34,32,24,19,38,26,17,29,30, 7,12,11,18, 5,28,14,23)$
19A1 $19^{2}$ $1332$ $19$ $36$ $( 1, 2, 9,13,27,29, 8,23,17,15,14,20, 4,33,26,35,38,24,34)( 3,25,31,30,28,22,37,16,18,32,36, 5, 6,11,21, 7,10,19,12)$
19A2 $19^{2}$ $1332$ $19$ $36$ $( 1, 9,27, 8,17,14, 4,26,38,34, 2,13,29,23,15,20,33,35,24)( 3,31,28,37,18,36, 6,21,10,12,25,30,22,16,32, 5,11, 7,19)$
19A3 $19^{2}$ $1332$ $19$ $36$ $( 1,13, 8,15, 4,35,34, 9,29,17,20,26,24, 2,27,23,14,33,38)( 3,30,37,32, 6, 7,12,31,22,18, 5,21,19,25,28,16,36,11,10)$
19A4 $19^{2}$ $1332$ $19$ $36$ $( 1,27,17, 4,38, 2,29,15,33,24, 9, 8,14,26,34,13,23,20,35)( 3,28,18, 6,10,25,22,32,11,19,31,37,36,21,12,30,16, 5, 7)$
19A5 $19^{2}$ $1332$ $19$ $36$ $( 1,29,14,35, 2, 8,20,38, 9,23, 4,24,13,17,33,34,27,15,26)( 3,22,36, 7,25,37, 5,10,31,16, 6,19,30,18,11,12,28,32,21)$
19A6 $19^{2}$ $1332$ $19$ $36$ $( 1, 8, 4,34,29,20,24,27,14,38,13,15,35, 9,17,26, 2,23,33)( 3,37, 6,12,22, 5,19,28,36,10,30,32, 7,31,18,21,25,16,11)$
19A7 $19^{2}$ $1332$ $19$ $36$ $( 1,23,26, 9,15,38,27,20,34, 8,33, 2,17,35,13,14,24,29, 4)( 3,16,21,31,32,10,28, 5,12,37,11,25,18, 7,30,36,19,22, 6)$
19A8 $19^{2}$ $1332$ $19$ $36$ $( 1,17,38,29,33, 9,14,34,23,35,27, 4, 2,15,24, 8,26,13,20)( 3,18,10,22,11,31,36,12,16, 7,28, 6,25,32,19,37,21,30, 5)$
19A9 $19^{2}$ $1332$ $19$ $36$ $( 1,15,34,17,24,23,38, 8,35,29,26,27,33,13, 4, 9,20, 2,14)( 3,32,12,18,19,16,10,37, 7,22,21,28,11,30, 6,31, 5,25,36)$
37A1 $37,1$ $684$ $37$ $36$ $( 1, 7,26,34,25,15,30,19, 2,27,24,21, 8,29,18,33,23,14,22, 3, 9,31, 4,38,16,20,13,36,37,11,12,35,28,32,10, 6,17)$
37A2 $37,1$ $684$ $37$ $36$ $( 1,26,25,30, 2,24, 8,18,23,22, 9, 4,16,13,37,12,28,10,17, 7,34,15,19,27,21,29,33,14, 3,31,38,20,36,11,35,32, 6)$

Malle's constant $a(G)$:     $1/18$

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 6A 9A1 9A2 9A4 18A1 18A5 18A7 19A1 19A2 19A3 19A4 19A5 19A6 19A7 19A8 19A9 37A1 37A2
Size 1 703 1406 1406 1406 1406 1406 1406 1406 1406 1332 1332 1332 1332 1332 1332 1332 1332 1332 684 684
2 P 1A 1A 3A 3A 9A2 9A4 9A1 9A1 9A4 9A2 19A2 19A4 19A6 19A8 19A9 19A7 19A5 19A3 19A1 37A2 37A1
3 P 1A 2A 1A 2A 3A 3A 3A 6A 6A 6A 19A3 19A6 19A9 19A7 19A4 19A1 19A2 19A5 19A8 37A1 37A2
19 P 1A 2A 3A 6A 9A1 9A2 9A4 18A1 18A5 18A7 1A 1A 1A 1A 1A 1A 1A 1A 1A 37A2 37A1
37 P 1A 2A 3A 6A 9A1 9A2 9A4 18A1 18A5 18A7 19A1 19A2 19A3 19A4 19A5 19A6 19A7 19A8 19A9 1A 1A
Type
25308.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
25308.a.19a1 R 19 −1 1 −1 1 1 1 −1 −1 −1 0 0 0 0 0 0 0 0 0 ζ37−18+ζ37−17+ζ37−15+ζ37−14+ζ37−13+ζ37−8+ζ37−6+ζ37−5+ζ37−2+1+ζ372+ζ375+ζ376+ζ378+ζ3713+ζ3714+ζ3715+ζ3717+ζ3718 −ζ37−18−ζ37−17−ζ37−15−ζ37−14−ζ37−13−ζ37−8−ζ37−6−ζ37−5−ζ37−2−ζ372−ζ375−ζ376−ζ378−ζ3713−ζ3714−ζ3715−ζ3717−ζ3718
25308.a.19a2 R 19 −1 1 −1 1 1 1 −1 −1 −1 0 0 0 0 0 0 0 0 0 −ζ37−18−ζ37−17−ζ37−15−ζ37−14−ζ37−13−ζ37−8−ζ37−6−ζ37−5−ζ37−2−ζ372−ζ375−ζ376−ζ378−ζ3713−ζ3714−ζ3715−ζ3717−ζ3718 ζ37−18+ζ37−17+ζ37−15+ζ37−14+ζ37−13+ζ37−8+ζ37−6+ζ37−5+ζ37−2+1+ζ372+ζ375+ζ376+ζ378+ζ3713+ζ3714+ζ3715+ζ3717+ζ3718
25308.a.36a1 R 36 0 0 0 0 0 0 0 0 0 −ζ19−1−ζ19 −ζ19−2−ζ192 −ζ19−3−ζ193 −ζ19−4−ζ194 −ζ19−5−ζ195 −ζ19−6−ζ196 −ζ19−7−ζ197 −ζ19−8−ζ198 −ζ19−9−ζ199 −1 −1
25308.a.36a2 R 36 0 0 0 0 0 0 0 0 0 −ζ19−2−ζ192 −ζ19−4−ζ194 −ζ19−6−ζ196 −ζ19−8−ζ198 −ζ19−9−ζ199 −ζ19−7−ζ197 −ζ19−5−ζ195 −ζ19−3−ζ193 −ζ19−1−ζ19 −1 −1
25308.a.36a3 R 36 0 0 0 0 0 0 0 0 0 −ζ19−3−ζ193 −ζ19−6−ζ196 −ζ19−9−ζ199 −ζ19−7−ζ197 −ζ19−4−ζ194 −ζ19−1−ζ19 −ζ19−2−ζ192 −ζ19−5−ζ195 −ζ19−8−ζ198 −1 −1
25308.a.36a4 R 36 0 0 0 0 0 0 0 0 0 −ζ19−4−ζ194 −ζ19−8−ζ198 −ζ19−7−ζ197 −ζ19−3−ζ193 −ζ19−1−ζ19 −ζ19−5−ζ195 −ζ19−9−ζ199 −ζ19−6−ζ196 −ζ19−2−ζ192 −1 −1
25308.a.36a5 R 36 0 0 0 0 0 0 0 0 0 −ζ19−5−ζ195 −ζ19−9−ζ199 −ζ19−4−ζ194 −ζ19−1−ζ19 −ζ19−6−ζ196 −ζ19−8−ζ198 −ζ19−3−ζ193 −ζ19−2−ζ192 −ζ19−7−ζ197 −1 −1
25308.a.36a6 R 36 0 0 0 0 0 0 0 0 0 −ζ19−6−ζ196 −ζ19−7−ζ197 −ζ19−1−ζ19 −ζ19−5−ζ195 −ζ19−8−ζ198 −ζ19−2−ζ192 −ζ19−4−ζ194 −ζ19−9−ζ199 −ζ19−3−ζ193 −1 −1
25308.a.36a7 R 36 0 0 0 0 0 0 0 0 0 −ζ19−7−ζ197 −ζ19−5−ζ195 −ζ19−2−ζ192 −ζ19−9−ζ199 −ζ19−3−ζ193 −ζ19−4−ζ194 −ζ19−8−ζ198 −ζ19−1−ζ19 −ζ19−6−ζ196 −1 −1
25308.a.36a8 R 36 0 0 0 0 0 0 0 0 0 −ζ19−8−ζ198 −ζ19−3−ζ193 −ζ19−5−ζ195 −ζ19−6−ζ196 −ζ19−2−ζ192 −ζ19−9−ζ199 −ζ19−1−ζ19 −ζ19−7−ζ197 −ζ19−4−ζ194 −1 −1
25308.a.36a9 R 36 0 0 0 0 0 0 0 0 0 −ζ19−9−ζ199 −ζ19−1−ζ19 −ζ19−8−ζ198 −ζ19−2−ζ192 −ζ19−7−ζ197 −ζ19−3−ζ193 −ζ19−6−ζ196 −ζ19−4−ζ194 −ζ19−5−ζ195 −1 −1
25308.a.37a R 37 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1 0 0
25308.a.38a R 38 2 2 2 −1 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 1 1
25308.a.38b R 38 −2 2 −2 −1 −1 −1 1 1 1 0 0 0 0 0 0 0 0 0 1 1
25308.a.38c1 R 38 2 −1 −1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−2+ζ92 ζ9−1+ζ9 ζ9−4+ζ94 0 0 0 0 0 0 0 0 0 1 1
25308.a.38c2 R 38 2 −1 −1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−1+ζ9 ζ9−4+ζ94 ζ9−2+ζ92 0 0 0 0 0 0 0 0 0 1 1
25308.a.38c3 R 38 2 −1 −1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−4+ζ94 ζ9−2+ζ92 ζ9−1+ζ9 0 0 0 0 0 0 0 0 0 1 1
25308.a.38d1 R 38 −2 −1 1 ζ9−4+ζ94 ζ9−1+ζ9 ζ9−2+ζ92 −ζ9−2−ζ92 −ζ9−1−ζ9 −ζ9−4−ζ94 0 0 0 0 0 0 0 0 0 1 1
25308.a.38d2 R 38 −2 −1 1 ζ9−2+ζ92 ζ9−4+ζ94 ζ9−1+ζ9 −ζ9−1−ζ9 −ζ9−4−ζ94 −ζ9−2−ζ92 0 0 0 0 0 0 0 0 0 1 1
25308.a.38d3 R 38 −2 −1 1 ζ9−1+ζ9 ζ9−2+ζ92 ζ9−4+ζ94 −ζ9−4−ζ94 −ζ9−2−ζ92 −ζ9−1−ζ9 0 0 0 0 0 0 0 0 0 1 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