Properties

Label 38T30
Degree $38$
Order $12996$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{19}:F_{19}$

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Show commands: Magma

magma: G := TransitiveGroup(38, 30);
 

Group invariants

Abstract group:  $D_{19}:F_{19}$
magma: IdentifyGroup(G);
 
Order:  $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$
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 $t$:  $30$
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:  $(1,6)(2,5)(3,4)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35)$, $(1,29,9,26,4,35,19,27,12,32,14,36,8,24,7,22,10,28)(2,31,6,20,13,34,11,30,17,23,18,25,15,38,5,37,16,21)(3,33)$
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
$9$:  $C_9$
$12$:  $C_6\times C_2$
$18$:  $C_{18}$ x 3
$36$:  36T2
$342$:  $F_{19}$ x 2
$684$:  38T9 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T30 x 8

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 $2^{19}$ $19$ $2$ $19$ $( 1,24)( 2,34)( 3,25)( 4,35)( 5,26)( 6,36)( 7,27)( 8,37)( 9,28)(10,38)(11,29)(12,20)(13,30)(14,21)(15,31)(16,22)(17,32)(18,23)(19,33)$
2B $2^{19}$ $19$ $2$ $19$ $( 1,36)( 2,26)( 3,35)( 4,25)( 5,34)( 6,24)( 7,33)( 8,23)( 9,32)(10,22)(11,31)(12,21)(13,30)(14,20)(15,29)(16,38)(17,28)(18,37)(19,27)$
2C $2^{18},1^{2}$ $361$ $2$ $18$ $( 1, 6)( 2, 5)( 3, 4)( 7,19)( 8,18)( 9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35)$
3A1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,10,16)( 2,17, 8)( 3, 5,19)( 4,12,11)( 6, 7,14)(13,18,15)(20,28,27)(21,35,38)(22,23,30)(24,37,33)(26,32,36)(29,34,31)$
3A-1 $3^{12},1^{2}$ $361$ $3$ $24$ $( 1,16,10)( 2, 8,17)( 3,19, 5)( 4,11,12)( 6,14, 7)(13,15,18)(20,27,28)(21,38,35)(22,30,23)(24,33,37)(26,36,32)(29,31,34)$
6A1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,16, 6,19, 4,14)( 2, 9,17,18,11, 3)( 5, 7,12,15,13, 8)(20,22,27,30,28,23)(21,34,38,29,35,31)(24,32,33,26,37,36)$
6A-1 $6^{6},2$ $361$ $6$ $31$ $( 1,29,11,31, 5,26)( 2,33,18,21,16,32)( 3,37, 6,30, 8,38)( 4,22,13,20,19,25)( 7,34,15,28,14,24)( 9,23,10,27,17,36)(12,35)$
6B1 $6^{6},1^{2}$ $361$ $6$ $30$ $( 1,10, 6,12, 3, 7)( 2,18,13,11,14,19)( 4,15, 8, 9,17, 5)(20,23,28,30,27,22)(21,31,35,29,38,34)(24,36,37,26,33,32)$
6B-1 $6^{6},2$ $361$ $6$ $31$ $( 1,32)( 2,38,12,22, 8,36)( 3,25, 4,31,15,21)( 5,37, 7,30,10,29)( 6,24,18,20,17,33)( 9,23,13,28,19,26)(11,35,16,27,14,34)$
6C1 $6^{6},2$ $361$ $6$ $31$ $( 1,37,19,24, 8,33)( 2,31,11,34,15,29)( 3,25)( 4,38,14,35,10,21)( 5,32, 6,26,17,36)( 7,20, 9,27,12,28)(13,22,18,30,16,23)$
6C-1 $6^{6},2$ $361$ $6$ $31$ $( 1,21,10,23,16,37)( 2,36,17,33, 8,31)( 3,32, 5,24,19,25)( 4,28,12,34,11,38)( 6,20, 7,35,14,26)( 9,27)(13,30,18,29,15,22)$
9A1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,14,13,16, 7,15,10, 6,18)( 2,11, 3, 8,12,19,17, 4, 5)(20,21,37,27,38,24,28,35,33)(22,34,36,30,29,32,23,31,26)$
9A-1 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 7,18,16, 6,13,10,14,15)( 2,12, 5, 8, 4, 3,17,11,19)(20,38,33,27,35,37,28,21,24)(22,29,26,30,31,36,23,34,32)$
9A2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,13, 7,10,18,14,16,15, 6)( 2, 3,12,17, 5,11, 8,19, 4)(20,37,38,28,33,21,27,24,35)(22,36,29,23,26,34,30,32,31)$
9A-2 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,15,14,10,13, 6,16,18, 7)( 2,19,11,17, 3, 4, 8, 5,12)(20,24,21,28,37,35,27,33,38)(22,32,34,23,36,31,30,26,29)$
9A4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1, 6,15,16,14,18,10, 7,13)( 2, 4,19, 8,11, 5,17,12, 3)(20,35,24,27,21,33,28,38,37)(22,31,32,30,34,26,23,29,36)$
9A-4 $9^{4},1^{2}$ $361$ $9$ $32$ $( 1,18, 6,10,15, 7,16,13,14)( 2, 5, 4,17,19,12, 8, 3,11)(20,33,35,28,24,38,27,37,21)(22,26,31,23,32,29,30,36,34)$
18A1 $18^{2},2$ $361$ $18$ $35$ $( 1,28, 6,32,15,24,16,21,14,27,18,34,10,20, 7,29,13,30)( 2,25, 4,38,19,31, 8,26,11,36, 5,35,17,37,12,33, 3,22)( 9,23)$
18A-1 $18^{2},2$ $361$ $18$ $35$ $( 1,34, 2,35, 6,20, 3,36,10,24,19,33,17,31, 9,23,15,29)( 4,37,14,28,16,30, 5,38,18,32,13,27,12,26, 8,22,11,25)( 7,21)$
18A5 $18^{2},2$ $361$ $18$ $35$ $( 1,35)( 2,34, 5,31,17,38, 8,28,10,26,18,37,12,24, 7,29, 6,30)( 3,33, 9,27,14,22,15,21,19,36,16,20, 4,32,13,23,11,25)$
18A-5 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,19, 5,18,10,12, 2,14,11, 7, 8, 3, 9,17,15, 6,13,16)(20,31,33,23,35,32,28,29,24,30,38,36,27,34,37,22,21,26)$
18A7 $18^{2},2$ $361$ $18$ $35$ $( 1,24,14,20,16,34, 9,23, 5,33,19,36, 8,35,18,29, 2,31)( 3,38, 7,28,12,25, 4,26,13,32,10,30,11,37,17,22,15,27)( 6,21)$
18A-7 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,12, 7,11, 4, 2,15,16,19, 9,17, 3,18, 6, 8,14,13,10)(20,29,37,23,38,26,28,34,33,30,21,32,27,31,24,22,35,36)$
18B1 $18^{2},2$ $361$ $18$ $35$ $( 1,29,18,20, 8,32,15,35,12,31,16,30,17,25, 3,38, 9,27)( 2,24, 4,33,14,21, 7,37,10,22, 6,23, 5,28,19,34,13,26)(11,36)$
18B-1 $18^{2},2$ $361$ $18$ $35$ $( 1,22, 5,34,16,29,13,20,19,38, 7,21,12,36, 2,25, 3,28)( 4,31,18,35, 9,27, 8,24,10,30, 6,37,14,23,17,32,11,33)(15,26)$
18B5 $18^{2},1^{2}$ $361$ $18$ $34$ $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(20,34,24,23,21,36,28,31,37,30,35,26,27,29,33,22,38,32)$
18B-5 $18^{2},2$ $361$ $18$ $35$ $( 1,21,19,35,14,29, 8,37,16,20,18,30, 9,23, 2,26, 5,22)( 3,31,10,28, 7,32,11,33,12,38,17,25, 4,36,15,34,13,24)( 6,27)$
18B7 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1, 8, 4, 9,17, 7,10,11, 5, 3,15,19,14, 6,16,13,12,18)(20,36,35,22,24,31,27,32,21,30,33,34,28,26,38,23,37,29)$
18B-7 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,11, 9,17, 4,18,19,15,12, 5,14,16, 8, 2, 7, 6,10,13)(20,26,21,22,37,34,27,36,38,30,24,29,28,32,35,23,33,31)$
18C1 $18^{2},2$ $361$ $18$ $35$ $( 1,20,14,29,17,34, 6,22, 2,28, 4,25, 3,36,13,21, 8,38)( 5,33,12,32,18,23,15,37, 7,30,11,24, 9,27,10,35,19,31)(16,26)$
18C-1 $18^{2},2$ $361$ $18$ $35$ $( 1,26,13,37, 9,27, 4,24,12,25, 3,31, 6,29, 5,36,18,21)( 2,38,19,33, 7,22,11,32,16,35, 8,34,17,28,14,30,15,23)(10,20)$
18C5 $18^{2},2$ $361$ $18$ $35$ $( 1,30, 5,38,12,33,10,29,16,22,17,24,14,37, 4,36,15,20)( 2,32)( 3,34,18,26,11,31,13,35, 7,23, 6,21, 9,27,19,28, 8,25)$
18C-5 $18^{2},1^{2}$ $361$ $18$ $34$ $( 1,13,19, 3,14,10, 8, 7,16,11,18,12, 9,17, 2, 4, 5,15)(20,32,38,22,33,29,27,26,35,30,37,31,28,36,21,23,24,34)$
18C7 $18^{2},2$ $361$ $18$ $35$ $( 1,20, 6,29,10,21,17,26,15,30, 2,37, 3,35,19,22, 9,23)( 4,33,16,28,18,24,12,36,11,38,14,32, 5,31,13,34, 8,25)( 7,27)$
18C-7 $18^{2},2$ $361$ $18$ $35$ $( 1,30,15,32, 8,31, 2,22, 5,36,13,29, 9,23,11,26,10,34)( 3,33,14,21,18,27,16,24,17,35, 7,20,12,37,19,38, 6,28)( 4,25)$
19A $19^{2}$ $18$ $19$ $36$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,22,24,26,28,30,32,34,36,38,21,23,25,27,29,31,33,35,37)$
19B $19^{2}$ $18$ $19$ $36$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$
19C $19^{2}$ $36$ $19$ $36$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,28,36,25,33,22,30,38,27,35,24,32,21,29,37,26,34,23,31)$
19D $19^{2}$ $36$ $19$ $36$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,29,38,28,37,27,36,26,35,25,34,24,33,23,32,22,31,21,30)$
19E $19^{2}$ $36$ $19$ $36$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$
19F $19^{2}$ $36$ $19$ $36$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19G $19^{2}$ $36$ $19$ $36$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$
19H $19^{2}$ $36$ $19$ $36$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
19I $19,1^{19}$ $36$ $19$ $18$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$
19J $19^{2}$ $36$ $19$ $36$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,26,29,32,35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
19K $19^{2}$ $36$ $19$ $36$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
38A $38$ $342$ $38$ $37$ $( 1,23,18,22,16,21,14,20,12,38,10,37, 8,36, 6,35, 4,34, 2,33,19,32,17,31,15,30,13,29,11,28, 9,27, 7,26, 5,25, 3,24)$
38B $38$ $342$ $38$ $37$ $( 1,27,19,37,18,28,17,38,16,29,15,20,14,30,13,21,12,31,11,22,10,32, 9,23, 8,33, 7,24, 6,34, 5,25, 4,35, 3,26, 2,36)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Character table

49 x 49 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed