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Properties

Label 38T6
38T6 1 3 1->3 11 1->11 2 4 2->4 12 2->12 25 3->25 26 4->26 5 5->1 37 5->37 6 6->2 38 6->38 7 15 7->15 35 7->35 8 16 8->16 36 8->36 9 29 9->29 33 9->33 10 30 10->30 34 10->34 11->6 31 11->31 12->5 32 12->32 13 20 13->20 13->29 14 19 14->19 14->30 28 15->28 15->33 27 16->27 16->34 17 17->9 17->25 18 18->10 18->26 23 19->23 19->23 24 20->24 20->24 21 22 21->22 21->38 22->37 23->13 24->14 25->28 26->27 27->3 28->4 29->18 30->17 31->32 33->8 34->7 35->22 36->21 37->36 38->35
Degree $38$
Order $228$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{38}:C_6$

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Copy content magma:G := TransitiveGroup(38, 6);
 

Group invariants

Abstract group:  $C_{38}:C_6$
Copy content magma:IdentifyGroup(G);
 
Order:  $228=2^{2} \cdot 3 \cdot 19$
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 $t$:  $6$
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:  $(1,11,6,2,12,5)(3,25,28,4,26,27)(7,15,33,8,16,34)(9,29,18,10,30,17)(13,20,24,14,19,23)(21,38,35,22,37,36)(31,32)$, $(1,3)(2,4)(5,37)(6,38)(7,35)(8,36)(9,33)(10,34)(11,31)(12,32)(13,29)(14,30)(15,28)(16,27)(17,25)(18,26)(19,23)(20,24)(21,22)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $C_6$ x 3
$12$:  $C_6\times C_2$
$114$:  $C_{19}:C_{6}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $C_{19}:C_{6}$

Low degree siblings

38T6

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 $2^{18},1^{2}$ $19$ $2$ $18$ $( 1, 6)( 2, 5)( 7,38)( 8,37)( 9,35)(10,36)(11,33)(12,34)(13,31)(14,32)(15,30)(16,29)(17,28)(18,27)(19,25)(20,26)(21,24)(22,23)$
2C $2^{19}$ $19$ $2$ $19$ $( 1,17)( 2,18)( 3,15)( 4,16)( 5,13)( 6,14)( 7,12)( 8,11)( 9,10)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,29)(28,30)$
3A1 $3^{12},1^{2}$ $19$ $3$ $24$ $( 1,27,19)( 2,28,20)( 5,17,26)( 6,18,25)( 7,32,10)( 8,31, 9)(11,22,16)(12,21,15)(13,35,37)(14,36,38)(23,29,33)(24,30,34)$
3A-1 $3^{12},1^{2}$ $19$ $3$ $24$ $( 1,19,27)( 2,20,28)( 5,26,17)( 6,25,18)( 7,10,32)( 8, 9,31)(11,16,22)(12,15,21)(13,37,35)(14,38,36)(23,33,29)(24,34,30)$
6A1 $6^{6},1^{2}$ $19$ $6$ $30$ $( 1,25,27, 6,19,18)( 2,26,28, 5,20,17)( 7,36,32,38,10,14)( 8,35,31,37, 9,13)(11,29,22,33,16,23)(12,30,21,34,15,24)$
6A-1 $6^{6},1^{2}$ $19$ $6$ $30$ $( 1,18,19, 6,27,25)( 2,17,20, 5,28,26)( 7,14,10,38,32,36)( 8,13, 9,37,31,35)(11,23,16,33,22,29)(12,24,15,34,21,30)$
6B1 $6^{6},2$ $19$ $6$ $31$ $( 1,22,30,17,35,28)( 2,21,29,18,36,27)( 3,37, 5,15,20,13)( 4,38, 6,16,19,14)( 7,31,33,12,26,24)( 8,32,34,11,25,23)( 9,10)$
6B-1 $6^{6},2$ $19$ $6$ $31$ $( 1,28,35,17,30,22)( 2,27,36,18,29,21)( 3,13,20,15, 5,37)( 4,14,19,16, 6,38)( 7,24,26,12,33,31)( 8,23,25,11,34,32)( 9,10)$
6C1 $6^{6},2$ $19$ $6$ $31$ $( 1,38,24, 2,37,23)( 3,13, 7, 4,14, 8)( 5,27,29, 6,28,30)( 9,17,35,10,18,36)(11,31,20,12,32,19)(15,22,25,16,21,26)(33,34)$
6C-1 $6^{6},2$ $19$ $6$ $31$ $( 1,23,37, 2,24,38)( 3, 8,14, 4, 7,13)( 5,30,28, 6,29,27)( 9,36,18,10,35,17)(11,19,32,12,20,31)(15,26,21,16,25,22)(33,34)$
19A1 $19^{2}$ $6$ $19$ $36$ $( 1,34,27,21,15, 9, 4,35,30,24,18,12, 6,37,31,25,19,13, 8)( 2,33,28,22,16,10, 3,36,29,23,17,11, 5,38,32,26,20,14, 7)$
19A2 $19^{2}$ $6$ $19$ $36$ $( 1,27,15, 4,30,18, 6,31,19, 8,34,21, 9,35,24,12,37,25,13)( 2,28,16, 3,29,17, 5,32,20, 7,33,22,10,36,23,11,38,26,14)$
19A4 $19^{2}$ $6$ $19$ $36$ $( 1,15,30, 6,19,34, 9,24,37,13,27, 4,18,31, 8,21,35,12,25)( 2,16,29, 5,20,33,10,23,38,14,28, 3,17,32, 7,22,36,11,26)$
38A1 $38$ $6$ $38$ $37$ $( 1,17,34,11,27, 5,21,38,15,32, 9,26, 4,20,35,14,30, 7,24, 2,18,33,12,28, 6,22,37,16,31,10,25, 3,19,36,13,29, 8,23)$
38A3 $38$ $6$ $38$ $37$ $( 1,11,21,32, 4,14,24,33, 6,16,25,36, 8,17,27,38, 9,20,30, 2,12,22,31, 3,13,23,34, 5,15,26,35, 7,18,28,37,10,19,29)$
38A9 $38$ $6$ $38$ $37$ $( 1,32,24,16, 8,38,30,22,13, 5,35,28,19,11, 4,33,25,17, 9, 2,31,23,15, 7,37,29,21,14, 6,36,27,20,12, 3,34,26,18,10)$

Malle's constant $a(G)$:     $1/18$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 19A1 19A2 19A4 38A1 38A3 38A9
Size 1 1 19 19 19 19 19 19 19 19 19 19 6 6 6 6 6 6
2 P 1A 1A 1A 1A 3A-1 3A1 3A1 3A-1 3A1 3A-1 3A-1 3A1 19A2 19A4 19A1 19A1 19A2 19A4
3 P 1A 2A 2B 2C 1A 1A 2B 2B 2C 2C 2A 2A 19A2 19A4 19A1 38A3 38A9 38A1
19 P 1A 2A 2B 2C 3A1 3A-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 1A 1A 1A 2A 2A 2A
Type
228.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
228.7.1b R 1 −1 −1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1
228.7.1c R 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 1 1 −1 −1 −1
228.7.1d R 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
228.7.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1
228.7.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1
228.7.1f1 C 1 −1 −1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 −1 −1 −1
228.7.1f2 C 1 −1 −1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 −1 −1 −1
228.7.1g1 C 1 −1 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 −1 −1 −1
228.7.1g2 C 1 −1 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 −1 −1 −1
228.7.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1
228.7.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1
228.7.6a1 R 6 6 0 0 0 0 0 0 0 0 0 0 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198
228.7.6a2 R 6 6 0 0 0 0 0 0 0 0 0 0 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195
228.7.6a3 R 6 6 0 0 0 0 0 0 0 0 0 0 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199
228.7.6b1 R 6 −6 0 0 0 0 0 0 0 0 0 0 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 −ζ19−5−ζ19−3−ζ19−2−ζ192−ζ193−ζ195 −ζ19−9−ζ19−6−ζ19−4−ζ194−ζ196−ζ199 −ζ19−8−ζ19−7−ζ19−1−ζ19−ζ197−ζ198
228.7.6b2 R 6 −6 0 0 0 0 0 0 0 0 0 0 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 −ζ19−9−ζ19−6−ζ19−4−ζ194−ζ196−ζ199 −ζ19−8−ζ19−7−ζ19−1−ζ19−ζ197−ζ198 −ζ19−5−ζ19−3−ζ19−2−ζ192−ζ193−ζ195
228.7.6b3 R 6 −6 0 0 0 0 0 0 0 0 0 0 ζ19−5+ζ19−3+ζ19−2+ζ192+ζ193+ζ195 ζ19−9+ζ19−6+ζ19−4+ζ194+ζ196+ζ199 ζ19−8+ζ19−7+ζ19−1+ζ19+ζ197+ζ198 −ζ19−8−ζ19−7−ζ19−1−ζ19−ζ197−ζ198 −ζ19−5−ζ19−3−ζ19−2−ζ192−ζ193−ζ195 −ζ19−9−ζ19−6−ζ19−4−ζ194−ζ196−ζ199

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed