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Properties

Label 38T3
38T3 1 30 1->30 33 1->33 2 29 2->29 34 2->34 3 31 3->31 32 3->32 4 4->31 4->32 5 5->29 5->34 6 6->30 6->33 7 27 7->27 36 7->36 8 28 8->28 35 8->35 9 25 9->25 37 9->37 10 26 10->26 38 10->38 11 11->2 23 11->23 12 12->1 24 12->24 13 13->3 21 13->21 14 14->4 22 14->22 15 15->6 19 15->19 16 16->5 20 16->20 17 17->7 18 18->8 19->10 20->9 21->11 22->12 23->13 24->14 25->16 26->15 27->17 28->18 29->20 30->19 31->22 32->21 33->24 34->23 35->26 35->38 36->25 36->37 37->27 38->28
Degree $38$
Order $76$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{38}$

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Copy content magma:G := TransitiveGroup(38, 3);
 

Group invariants

Abstract group:  $D_{38}$
Copy content magma:IdentifyGroup(G);
 
Order:  $76=2^{2} \cdot 19$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $38$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,33)(2,34)(3,32)(4,31)(5,29)(6,30)(7,27)(8,28)(9,25)(10,26)(11,23)(12,24)(13,21)(14,22)(15,19)(16,20)(35,38)(36,37)$, $(1,30,19,10,38,28,18,8,35,26,15,6,33,24,14,4,32,21,11,2,29,20,9,37,27,17,7,36,25,16,5,34,23,13,3,31,22,12)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$38$:  $D_{19}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $D_{19}$

Low degree siblings

38T3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{38}$ $1$ $1$ $0$ $()$
2A $2^{19}$ $1$ $2$ $19$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)$
2B $2^{18},1^{2}$ $19$ $2$ $18$ $( 3,38)( 4,37)( 5,35)( 6,36)( 7,33)( 8,34)( 9,32)(10,31)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26)(17,24)(18,23)(19,22)(20,21)$
2C $2^{19}$ $19$ $2$ $19$ $( 1,21)( 2,22)( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)(11,12)(23,37)(24,38)(25,36)(26,35)(27,34)(28,33)(29,31)(30,32)$
19A1 $19^{2}$ $2$ $19$ $36$ $( 1, 3, 5, 7, 9,11,14,15,18,19,22,23,25,27,29,32,33,35,38)( 2, 4, 6, 8,10,12,13,16,17,20,21,24,26,28,30,31,34,36,37)$
19A2 $19^{2}$ $2$ $19$ $36$ $( 1, 5, 9,14,18,22,25,29,33,38, 3, 7,11,15,19,23,27,32,35)( 2, 6,10,13,17,21,26,30,34,37, 4, 8,12,16,20,24,28,31,36)$
19A3 $19^{2}$ $2$ $19$ $36$ $( 1, 7,14,19,25,32,38, 5,11,18,23,29,35, 3, 9,15,22,27,33)( 2, 8,13,20,26,31,37, 6,12,17,24,30,36, 4,10,16,21,28,34)$
19A4 $19^{2}$ $2$ $19$ $36$ $( 1, 9,18,25,33, 3,11,19,27,35, 5,14,22,29,38, 7,15,23,32)( 2,10,17,26,34, 4,12,20,28,36, 6,13,21,30,37, 8,16,24,31)$
19A5 $19^{2}$ $2$ $19$ $36$ $( 1,11,22,32, 3,14,23,33, 5,15,25,35, 7,18,27,38, 9,19,29)( 2,12,21,31, 4,13,24,34, 6,16,26,36, 8,17,28,37,10,20,30)$
19A6 $19^{2}$ $2$ $19$ $36$ $( 1,14,25,38,11,23,35, 9,22,33, 7,19,32, 5,18,29, 3,15,27)( 2,13,26,37,12,24,36,10,21,34, 8,20,31, 6,17,30, 4,16,28)$
19A7 $19^{2}$ $2$ $19$ $36$ $( 1,15,29, 5,19,33, 9,23,38,14,27, 3,18,32, 7,22,35,11,25)( 2,16,30, 6,20,34,10,24,37,13,28, 4,17,31, 8,21,36,12,26)$
19A8 $19^{2}$ $2$ $19$ $36$ $( 1,18,33,11,27, 5,22,38,15,32, 9,25, 3,19,35,14,29, 7,23)( 2,17,34,12,28, 6,21,37,16,31,10,26, 4,20,36,13,30, 8,24)$
19A9 $19^{2}$ $2$ $19$ $36$ $( 1,19,38,18,35,15,33,14,32,11,29, 9,27, 7,25, 5,23, 3,22)( 2,20,37,17,36,16,34,13,31,12,30,10,28, 8,26, 6,24, 4,21)$
38A1 $38$ $2$ $38$ $37$ $( 1,21, 3,24, 5,26, 7,28, 9,30,11,31,14,34,15,36,18,37,19, 2,22, 4,23, 6,25, 8,27,10,29,12,32,13,33,16,35,17,38,20)$
38A3 $38$ $2$ $38$ $37$ $( 1,24, 7,30,14,36,19, 4,25,10,32,16,38,21, 5,28,11,34,18, 2,23, 8,29,13,35,20, 3,26, 9,31,15,37,22, 6,27,12,33,17)$
38A5 $38$ $2$ $38$ $37$ $( 1,26,11,36,22, 8,32,17, 3,28,14,37,23,10,33,20, 5,30,15, 2,25,12,35,21, 7,31,18, 4,27,13,38,24, 9,34,19, 6,29,16)$
38A7 $38$ $2$ $38$ $37$ $( 1,28,15, 4,29,17, 5,31,19, 8,33,21, 9,36,23,12,38,26,14, 2,27,16, 3,30,18, 6,32,20, 7,34,22,10,35,24,11,37,25,13)$
38A9 $38$ $2$ $38$ $37$ $( 1,30,19,10,38,28,18, 8,35,26,15, 6,33,24,14, 4,32,21,11, 2,29,20, 9,37,27,17, 7,36,25,16, 5,34,23,13, 3,31,22,12)$
38A11 $38$ $2$ $38$ $37$ $( 1,31,23,16, 7,37,29,21,14, 6,35,28,19,12, 3,34,25,17, 9, 2,32,24,15, 8,38,30,22,13, 5,36,27,20,11, 4,33,26,18,10)$
38A13 $38$ $2$ $38$ $37$ $( 1,34,27,21,15,10, 3,36,29,24,18,12, 5,37,32,26,19,13, 7, 2,33,28,22,16, 9, 4,35,30,23,17,11, 6,38,31,25,20,14, 8)$
38A15 $38$ $2$ $38$ $37$ $( 1,36,32,28,23,20,15,12, 7, 4,38,34,29,26,22,17,14,10, 5, 2,35,31,27,24,19,16,11, 8, 3,37,33,30,25,21,18,13, 9, 6)$
38A17 $38$ $2$ $38$ $37$ $( 1,37,35,34,32,30,27,26,23,21,19,17,15,13,11,10, 7, 6, 3, 2,38,36,33,31,29,28,25,24,22,20,18,16,14,12, 9, 8, 5, 4)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 19A1 19A2 19A3 19A4 19A5 19A6 19A7 19A8 19A9 38A1 38A3 38A5 38A7 38A9 38A11 38A13 38A15 38A17
Size 1 1 19 19 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 19A2 19A4 19A6 19A8 19A9 19A7 19A5 19A3 19A1 19A1 19A3 19A5 19A7 19A9 19A8 19A6 19A4 19A2
19 P 1A 2A 2B 2C 19A3 19A6 19A9 19A7 19A4 19A1 19A2 19A5 19A8 38A3 38A9 38A15 38A17 38A11 38A5 38A1 38A7 38A13
Type
76.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
76.3.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1
76.3.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −1
76.3.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
76.3.2a1 R 2 2 0 0 ζ19−9+ζ199 ζ19−6+ζ196 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−7+ζ197 ζ19−3+ζ193 ζ19−1+ζ19 ζ19−5+ζ195 ζ19−7+ζ197 ζ19−8+ζ198 ζ19−4+ζ194 ζ19−2+ζ192 ζ19−6+ζ196 ζ19−9+ζ199 ζ19−5+ζ195 ζ19−1+ζ19 ζ19−3+ζ193
76.3.2a2 R 2 2 0 0 ζ19−8+ζ198 ζ19−1+ζ19 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−2+ζ192 ζ19−9+ζ199 ζ19−3+ζ193 ζ19−4+ζ194 ζ19−2+ζ192 ζ19−5+ζ195 ζ19−7+ζ197 ζ19−6+ζ196 ζ19−1+ζ19 ζ19−8+ζ198 ζ19−4+ζ194 ζ19−3+ζ193 ζ19−9+ζ199
76.3.2a3 R 2 2 0 0 ζ19−7+ζ197 ζ19−8+ζ198 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−3+ζ193 ζ19−4+ζ194 ζ19−5+ζ195 ζ19−6+ζ196 ζ19−3+ζ193 ζ19−2+ζ192 ζ19−1+ζ19 ζ19−9+ζ199 ζ19−8+ζ198 ζ19−7+ζ197 ζ19−6+ζ196 ζ19−5+ζ195 ζ19−4+ζ194
76.3.2a4 R 2 2 0 0 ζ19−6+ζ196 ζ19−4+ζ194 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−8+ζ198 ζ19−2+ζ192 ζ19−7+ζ197 ζ19−3+ζ193 ζ19−8+ζ198 ζ19−1+ζ19 ζ19−9+ζ199 ζ19−5+ζ195 ζ19−4+ζ194 ζ19−6+ζ196 ζ19−3+ζ193 ζ19−7+ζ197 ζ19−2+ζ192
76.3.2a5 R 2 2 0 0 ζ19−5+ζ195 ζ19−3+ζ193 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−6+ζ196 ζ19−8+ζ198 ζ19−9+ζ199 ζ19−7+ζ197 ζ19−6+ζ196 ζ19−4+ζ194 ζ19−2+ζ192 ζ19−1+ζ19 ζ19−3+ζ193 ζ19−5+ζ195 ζ19−7+ζ197 ζ19−9+ζ199 ζ19−8+ζ198
76.3.2a6 R 2 2 0 0 ζ19−4+ζ194 ζ19−9+ζ199 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−1+ζ19 ζ19−5+ζ195 ζ19−8+ζ198 ζ19−2+ζ192 ζ19−1+ζ19 ζ19−7+ζ197 ζ19−6+ζ196 ζ19−3+ζ193 ζ19−9+ζ199 ζ19−4+ζ194 ζ19−2+ζ192 ζ19−8+ζ198 ζ19−5+ζ195
76.3.2a7 R 2 2 0 0 ζ19−3+ζ193 ζ19−2+ζ192 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−4+ζ194 ζ19−1+ζ19 ζ19−6+ζ196 ζ19−8+ζ198 ζ19−4+ζ194 ζ19−9+ζ199 ζ19−5+ζ195 ζ19−7+ζ197 ζ19−2+ζ192 ζ19−3+ζ193 ζ19−8+ζ198 ζ19−6+ζ196 ζ19−1+ζ19
76.3.2a8 R 2 2 0 0 ζ19−2+ζ192 ζ19−5+ζ195 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−9+ζ199 ζ19−7+ζ197 ζ19−4+ζ194 ζ19−1+ζ19 ζ19−9+ζ199 ζ19−6+ζ196 ζ19−3+ζ193 ζ19−8+ζ198 ζ19−5+ζ195 ζ19−2+ζ192 ζ19−1+ζ19 ζ19−4+ζ194 ζ19−7+ζ197
76.3.2a9 R 2 2 0 0 ζ19−1+ζ19 ζ19−7+ζ197 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−5+ζ195 ζ19−6+ζ196 ζ19−2+ζ192 ζ19−9+ζ199 ζ19−5+ζ195 ζ19−3+ζ193 ζ19−8+ζ198 ζ19−4+ζ194 ζ19−7+ζ197 ζ19−1+ζ19 ζ19−9+ζ199 ζ19−2+ζ192 ζ19−6+ζ196
76.3.2b1 R 2 −2 0 0 ζ19−9+ζ199 ζ19−6+ζ196 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−7+ζ197 ζ19−3+ζ193 ζ19−1+ζ19 ζ19−5+ζ195 −ζ19−7−ζ197 −ζ19−8−ζ198 −ζ19−4−ζ194 −ζ19−2−ζ192 −ζ19−6−ζ196 −ζ19−9−ζ199 −ζ19−5−ζ195 −ζ19−1−ζ19 −ζ19−3−ζ193
76.3.2b2 R 2 −2 0 0 ζ19−8+ζ198 ζ19−1+ζ19 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−2+ζ192 ζ19−9+ζ199 ζ19−3+ζ193 ζ19−4+ζ194 −ζ19−2−ζ192 −ζ19−5−ζ195 −ζ19−7−ζ197 −ζ19−6−ζ196 −ζ19−1−ζ19 −ζ19−8−ζ198 −ζ19−4−ζ194 −ζ19−3−ζ193 −ζ19−9−ζ199
76.3.2b3 R 2 −2 0 0 ζ19−7+ζ197 ζ19−8+ζ198 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−3+ζ193 ζ19−4+ζ194 ζ19−5+ζ195 ζ19−6+ζ196 −ζ19−3−ζ193 −ζ19−2−ζ192 −ζ19−1−ζ19 −ζ19−9−ζ199 −ζ19−8−ζ198 −ζ19−7−ζ197 −ζ19−6−ζ196 −ζ19−5−ζ195 −ζ19−4−ζ194
76.3.2b4 R 2 −2 0 0 ζ19−6+ζ196 ζ19−4+ζ194 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−1+ζ19 ζ19−8+ζ198 ζ19−2+ζ192 ζ19−7+ζ197 ζ19−3+ζ193 −ζ19−8−ζ198 −ζ19−1−ζ19 −ζ19−9−ζ199 −ζ19−5−ζ195 −ζ19−4−ζ194 −ζ19−6−ζ196 −ζ19−3−ζ193 −ζ19−7−ζ197 −ζ19−2−ζ192
76.3.2b5 R 2 −2 0 0 ζ19−5+ζ195 ζ19−3+ζ193 ζ19−1+ζ19 ζ19−2+ζ192 ζ19−4+ζ194 ζ19−6+ζ196 ζ19−8+ζ198 ζ19−9+ζ199 ζ19−7+ζ197 −ζ19−6−ζ196 −ζ19−4−ζ194 −ζ19−2−ζ192 −ζ19−1−ζ19 −ζ19−3−ζ193 −ζ19−5−ζ195 −ζ19−7−ζ197 −ζ19−9−ζ199 −ζ19−8−ζ198
76.3.2b6 R 2 −2 0 0 ζ19−4+ζ194 ζ19−9+ζ199 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−7+ζ197 ζ19−1+ζ19 ζ19−5+ζ195 ζ19−8+ζ198 ζ19−2+ζ192 −ζ19−1−ζ19 −ζ19−7−ζ197 −ζ19−6−ζ196 −ζ19−3−ζ193 −ζ19−9−ζ199 −ζ19−4−ζ194 −ζ19−2−ζ192 −ζ19−8−ζ198 −ζ19−5−ζ195
76.3.2b7 R 2 −2 0 0 ζ19−3+ζ193 ζ19−2+ζ192 ζ19−7+ζ197 ζ19−5+ζ195 ζ19−9+ζ199 ζ19−4+ζ194 ζ19−1+ζ19 ζ19−6+ζ196 ζ19−8+ζ198 −ζ19−4−ζ194 −ζ19−9−ζ199 −ζ19−5−ζ195 −ζ19−7−ζ197 −ζ19−2−ζ192 −ζ19−3−ζ193 −ζ19−8−ζ198 −ζ19−6−ζ196 −ζ19−1−ζ19
76.3.2b8 R 2 −2 0 0 ζ19−2+ζ192 ζ19−5+ζ195 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−6+ζ196 ζ19−9+ζ199 ζ19−7+ζ197 ζ19−4+ζ194 ζ19−1+ζ19 −ζ19−9−ζ199 −ζ19−6−ζ196 −ζ19−3−ζ193 −ζ19−8−ζ198 −ζ19−5−ζ195 −ζ19−2−ζ192 −ζ19−1−ζ19 −ζ19−4−ζ194 −ζ19−7−ζ197
76.3.2b9 R 2 −2 0 0 ζ19−1+ζ19 ζ19−7+ζ197 ζ19−4+ζ194 ζ19−8+ζ198 ζ19−3+ζ193 ζ19−5+ζ195 ζ19−6+ζ196 ζ19−2+ζ192 ζ19−9+ζ199 −ζ19−5−ζ195 −ζ19−3−ζ193 −ζ19−8−ζ198 −ζ19−4−ζ194 −ζ19−7−ζ197 −ζ19−1−ζ19 −ζ19−9−ζ199 −ζ19−2−ζ192 −ζ19−6−ζ196

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed