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Properties

Label 38T8
Degree $38$
Order $342$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{19}$

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Show commands: Magma

magma: G := TransitiveGroup(38, 8);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$9$:  $C_9$
$18$:  $C_{18}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $F_{19}$

Low degree siblings

19T6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{38}$ $1$ $1$ $()$
$9^{4},1^{2}$ $19$ $9$ $( 3, 9,34,15,19,35,24,13,11)( 4,10,33,16,20,36,23,14,12)( 5,18,28,30,37,32, 7, 26,22)( 6,17,27,29,38,31, 8,25,21)$
$9^{4},1^{2}$ $19$ $9$ $( 3,11,13,24,35,19,15,34, 9)( 4,12,14,23,36,20,16,33,10)( 5,22,26, 7,32,37,30, 28,18)( 6,21,25, 8,31,38,29,27,17)$
$9^{4},1^{2}$ $19$ $9$ $( 3,13,35,15, 9,11,24,19,34)( 4,14,36,16,10,12,23,20,33)( 5,26,32,30,18,22, 7, 37,28)( 6,25,31,29,17,21, 8,38,27)$
$3^{12},1^{2}$ $19$ $3$ $( 3,15,24)( 4,16,23)( 5,30, 7)( 6,29, 8)( 9,19,13)(10,20,14)(11,34,35) (12,33,36)(17,38,25)(18,37,26)(21,27,31)(22,28,32)$
$9^{4},1^{2}$ $19$ $9$ $( 3,19,11,15,13,34,24, 9,35)( 4,20,12,16,14,33,23,10,36)( 5,37,22,30,26,28, 7, 18,32)( 6,38,21,29,25,27, 8,17,31)$
$3^{12},1^{2}$ $19$ $3$ $( 3,24,15)( 4,23,16)( 5, 7,30)( 6, 8,29)( 9,13,19)(10,14,20)(11,35,34) (12,36,33)(17,25,38)(18,26,37)(21,31,27)(22,32,28)$
$9^{4},1^{2}$ $19$ $9$ $( 3,34,19,24,11, 9,15,35,13)( 4,33,20,23,12,10,16,36,14)( 5,28,37, 7,22,18,30, 32,26)( 6,27,38, 8,21,17,29,31,25)$
$9^{4},1^{2}$ $19$ $9$ $( 3,35, 9,24,34,13,15,11,19)( 4,36,10,23,33,14,16,12,20)( 5,32,18, 7,28,26,30, 22,37)( 6,31,17, 8,27,25,29,21,38)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3, 5, 9,18,34,28,15,30,19,37,35,32,24, 7,13,26,11,22)( 4, 6,10,17,33, 27,16,29,20,38,36,31,23, 8,14,25,12,21)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3, 7,19,18,11,32,15, 5,13,37,34,22,24,30, 9,26,35,28)( 4, 8,20,17,12, 31,16, 6,14,38,33,21,23,29,10,25,36,27)$
$6^{6},2$ $19$ $6$ $( 1, 2)( 3,18,15,37,24,26)( 4,17,16,38,23,25)( 5,34,30,35, 7,11) ( 6,33,29,36, 8,12)( 9,28,19,32,13,22)(10,27,20,31,14,21)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3,22,11,26,13, 7,24,32,35,37,19,30,15,28,34,18, 9, 5)( 4,21,12,25,14, 8,23,31,36,38,20,29,16,27,33,17,10, 6)$
$6^{6},2$ $19$ $6$ $( 1, 2)( 3,26,24,37,15,18)( 4,25,23,38,16,17)( 5,11, 7,35,30,34) ( 6,12, 8,36,29,33)( 9,22,13,32,19,28)(10,21,14,31,20,27)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3,28,35,26, 9,30,24,22,34,37,13, 5,15,32,11,18,19, 7)( 4,27,36,25,10, 29,23,21,33,38,14, 6,16,31,12,17,20, 8)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3,30,13,18,35,22,15, 7, 9,37,11,28,24, 5,19,26,34,32)( 4,29,14,17,36, 21,16, 8,10,38,12,27,23, 6,20,25,33,31)$
$18^{2},2$ $19$ $18$ $( 1, 2)( 3,32,34,26,19, 5,24,28,11,37, 9, 7,15,22,35,18,13,30)( 4,31,33,25,20, 6,23,27,12,38,10, 8,16,21,36,17,14,29)$
$2^{19}$ $19$ $2$ $( 1, 2)( 3,37)( 4,38)( 5,35)( 6,36)( 7,34)( 8,33)( 9,32)(10,31)(11,30)(12,29) (13,28)(14,27)(15,26)(16,25)(17,23)(18,24)(19,22)(20,21)$
$19^{2}$ $18$ $19$ $( 1, 3, 6, 8, 9,11,13,15,17,19,21,24,25,27,29,31,34,35,38)( 2, 4, 5, 7,10,12, 14,16,18,20,22,23,26,28,30,32,33,36,37)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $342=2 \cdot 3^{2} \cdot 19$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  342.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 9A1 9A-1 9A2 9A-2 9A4 9A-4 18A1 18A-1 18A5 18A-5 18A7 18A-7 19A
Size 1 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 18
2 P 1A 1A 3A-1 3A1 3A1 3A-1 9A-2 9A-4 9A4 9A2 9A-1 9A1 9A2 9A-2 9A-4 9A-1 9A1 9A4 19A
3 P 1A 2A 1A 1A 2A 2A 3A-1 3A1 3A-1 3A1 3A1 3A-1 6A-1 6A1 6A-1 6A-1 6A1 6A1 19A
19 P 1A 2A 3A1 3A-1 6A1 6A-1 9A-1 9A-2 9A2 9A1 9A4 9A-4 18A-7 18A7 18A5 18A-1 18A1 18A-5 1A
Type
342.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
342.7.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
342.7.1c1 C 1 1 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 1
342.7.1c2 C 1 1 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 1
342.7.1d1 C 1 −1 1 1 −1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1
342.7.1d2 C 1 −1 1 1 −1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1
342.7.1e1 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 ζ9−2 ζ92 ζ9−1 ζ9 ζ94 ζ9−4 1
342.7.1e2 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 ζ92 ζ9−2 ζ9 ζ9−1 ζ9−4 ζ94 1
342.7.1e3 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 ζ9 ζ9−1 ζ9−4 ζ94 ζ9−2 ζ92 1
342.7.1e4 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 ζ9−1 ζ9 ζ94 ζ9−4 ζ92 ζ9−2 1
342.7.1e5 C 1 1 ζ9−3 ζ93 ζ93 ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 ζ94 ζ9−4 ζ92 ζ9−2 ζ9 ζ9−1 1
342.7.1e6 C 1 1 ζ93 ζ9−3 ζ9−3 ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−4 ζ94 ζ9−2 ζ92 ζ9−1 ζ9 1
342.7.1f1 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−4 ζ94 ζ9 ζ9−1 ζ92 ζ9−2 −ζ9−2 −ζ92 −ζ9−1 −ζ9 −ζ94 −ζ9−4 1
342.7.1f2 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ94 ζ9−4 ζ9−1 ζ9 ζ9−2 ζ92 −ζ92 −ζ9−2 −ζ9 −ζ9−1 −ζ9−4 −ζ94 1
342.7.1f3 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ92 ζ9−2 ζ94 ζ9−4 ζ9−1 ζ9 −ζ9 −ζ9−1 −ζ9−4 −ζ94 −ζ9−2 −ζ92 1
342.7.1f4 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9−2 ζ92 ζ9−4 ζ94 ζ9 ζ9−1 −ζ9−1 −ζ9 −ζ94 −ζ9−4 −ζ92 −ζ9−2 1
342.7.1f5 C 1 −1 ζ9−3 ζ93 −ζ93 −ζ9−3 ζ9−1 ζ9 ζ9−2 ζ92 ζ9−4 ζ94 −ζ94 −ζ9−4 −ζ92 −ζ9−2 −ζ9 −ζ9−1 1
342.7.1f6 C 1 −1 ζ93 ζ9−3 −ζ9−3 −ζ93 ζ9 ζ9−1 ζ92 ζ9−2 ζ94 ζ9−4 −ζ9−4 −ζ94 −ζ9−2 −ζ92 −ζ9−1 −ζ9 1
342.7.18a R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1

magma: CharacterTable(G);