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Properties

Label 38T18
Degree $38$
Order $4332$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{19}^2:C_{12}$

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Show commands: Magma

magma: G := TransitiveGroup(38, 18);
 

Group invariants

Abstract group:  $C_{19}^2:C_{12}$
magma: IdentifyGroup(G);
 
Order:  $4332=2^{2} \cdot 3 \cdot 19^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T18 x 9

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 $2^{18},1^{2}$ $361$ $2$ $18$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,19)(12,18)(13,17)(14,16)(20,26)(21,25)(22,24)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)$
3A1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,13,12)( 2, 5,19)( 3,16, 7)( 4, 8,14)( 6,11, 9)(10,17,18)(20,28,21)(22,31,35)(24,34,30)(25,26,37)(27,29,32)(33,38,36)$
3A-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,12,13)( 2,19, 5)( 3, 7,16)( 4,14, 8)( 6, 9,11)(10,18,17)(20,21,28)(22,35,31)(24,30,34)(25,37,26)(27,32,29)(33,36,38)$
4A1 $4^{9},2$ $361$ $4$ $28$ $( 1,38,10,27)( 2,22, 9,24)( 3,25, 8,21)( 4,28, 7,37)( 5,31, 6,34)(11,30,19,35)(12,33,18,32)(13,36,17,29)(14,20,16,26)(15,23)$
4A-1 $4^{9},2$ $361$ $4$ $28$ $( 1,27,10,38)( 2,24, 9,22)( 3,21, 8,25)( 4,37, 7,28)( 5,34, 6,31)(11,35,19,30)(12,32,18,33)(13,29,17,36)(14,26,16,20)(15,23)$
6A1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,18,13,10,12,17)( 2,11, 5, 9,19, 6)( 3, 4,16, 8, 7,14)(20,25,28,26,21,37)(22,30,31,24,35,34)(27,33,29,38,32,36)$
6A-1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,17,12,10,13,18)( 2, 6,19, 9, 5,11)( 3,14, 7, 8,16, 4)(20,37,21,26,28,25)(22,34,35,24,31,30)(27,36,32,38,29,33)$
12A1 $12^{3},2$ $361$ $12$ $34$ $( 1,29,18,38,13,32,10,36,12,27,17,33)( 2,34,11,22, 5,30, 9,31,19,24, 6,35)( 3,20, 4,25,16,28, 8,26, 7,21,14,37)(15,23)$
12A-1 $12^{3},2$ $361$ $12$ $34$ $( 1,33,17,27,12,36,10,32,13,38,18,29)( 2,35, 6,24,19,31, 9,30, 5,22,11,34)( 3,37,14,21, 7,26, 8,28,16,25, 4,20)(15,23)$
12A5 $12^{3},2$ $361$ $12$ $34$ $( 1,32,17,38,12,29,10,33,13,27,18,36)( 2,30, 6,22,19,34, 9,35, 5,24,11,31)( 3,28,14,25, 7,20, 8,37,16,21, 4,26)(15,23)$
12A-5 $12^{3},2$ $361$ $12$ $34$ $( 1,36,18,27,13,33,10,29,12,38,17,32)( 2,31,11,24, 5,35, 9,34,19,22, 6,30)( 3,26, 4,21,16,37, 8,20, 7,25,14,28)(15,23)$
19A1 $19^{2}$ $12$ $19$ $36$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19A2 $19^{2}$ $12$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19A4 $19^{2}$ $12$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$
19B1 $19^{2}$ $12$ $19$ $36$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
19B2 $19^{2}$ $12$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,26,32,38,25,31,37,24,30,36,23,29,35,22,28,34,21,27,33)$
19B4 $19^{2}$ $12$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$
19C1 $19^{2}$ $12$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19C2 $19^{2}$ $12$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19C4 $19^{2}$ $12$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
19D1 $19^{2}$ $12$ $19$ $36$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,37,35,33,31,29,27,25,23,21,38,36,34,32,30,28,26,24,22)$
19D2 $19^{2}$ $12$ $19$ $36$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
19D4 $19^{2}$ $12$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19E1 $19^{2}$ $12$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
19E2 $19^{2}$ $12$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19E4 $19^{2}$ $12$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$
19F1 $19^{2}$ $12$ $19$ $36$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,36,33,30,27,24,21,37,34,31,28,25,22,38,35,32,29,26,23)$
19F2 $19^{2}$ $12$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$
19F4 $19^{2}$ $12$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
19G1 $19^{2}$ $12$ $19$ $36$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$
19G2 $19^{2}$ $12$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$
19G4 $19^{2}$ $12$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$
19H1 $19^{2}$ $12$ $19$ $36$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
19H2 $19^{2}$ $12$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19H4 $19^{2}$ $12$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19I1 $19,1^{19}$ $12$ $19$ $18$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)$
19I2 $19,1^{19}$ $12$ $19$ $18$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)$
19I4 $19,1^{19}$ $12$ $19$ $18$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)$
19J1 $19^{2}$ $12$ $19$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19J2 $19^{2}$ $12$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,35,31,27,23,38,34,30,26,22,37,33,29,25,21,36,32,28,24)$
19J4 $19^{2}$ $12$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Character table

42 x 42 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed