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Properties

Label 37T8
Degree $37$
Order $666$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{37}:C_{18}$

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Show commands: Magma

magma: G := TransitiveGroup(37, 8);
 

Group action invariants

Degree $n$:  $37$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{37}:C_{18}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4,16,27,34,25,26,30,9,36,33,21,10,3,12,11,7,28)(2,8,32,17,31,13,15,23,18,35,29,5,20,6,24,22,14,19), (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)
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

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $666=2 \cdot 3^{2} \cdot 37$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  666.7
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 37A1 37A2
Size 1 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 18 18
2 P 1A 1A 3A-1 3A1 3A1 3A-1 9A-2 9A-4 9A4 9A2 9A-1 9A1 9A2 9A-1 9A-2 9A-4 9A1 9A4 37A2 37A1
3 P 1A 2A 1A 1A 2A 2A 3A-1 3A1 3A-1 3A1 3A1 3A-1 6A-1 6A-1 6A1 6A-1 6A1 6A1 37A1 37A2
37 P 1A 2A 3A1 3A-1 6A1 6A-1 9A-1 9A-2 9A2 9A1 9A4 9A-4 18A-7 18A-1 18A7 18A5 18A1 18A-5 1A 1A
Type
666.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
666.7.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1
666.7.1c1 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 1
666.7.1c2 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 1
666.7.1d1 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 1
666.7.1d2 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 1
666.7.1e1 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 1
666.7.1e2 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 1
666.7.1e3 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 1
666.7.1e4 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 1
666.7.1e5 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 1
666.7.1e6 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 1
666.7.1f1 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 1
666.7.1f2 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 1
666.7.1f3 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 1
666.7.1f4 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 1
666.7.1f5 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 1
666.7.1f6 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 1
666.7.18a1 R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ζ37−18+ζ37−17+ζ37−15+ζ37−14+ζ37−13+ζ37−8+ζ37−6+ζ37−5+ζ37−2+ζ372+ζ375+ζ376+ζ378+ζ3713+ζ3714+ζ3715+ζ3717+ζ3718 −ζ37−18−ζ37−17−ζ37−15−ζ37−14−ζ37−13−ζ37−8−ζ37−6−ζ37−5−ζ37−2−1−ζ372−ζ375−ζ376−ζ378−ζ3713−ζ3714−ζ3715−ζ3717−ζ3718
666.7.18a2 R 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −ζ37−18−ζ37−17−ζ37−15−ζ37−14−ζ37−13−ζ37−8−ζ37−6−ζ37−5−ζ37−2−1−ζ372−ζ375−ζ376−ζ378−ζ3713−ζ3714−ζ3715−ζ3717−ζ3718 ζ37−18+ζ37−17+ζ37−15+ζ37−14+ζ37−13+ζ37−8+ζ37−6+ζ37−5+ζ37−2+ζ372+ζ375+ζ376+ζ378+ζ3713+ζ3714+ζ3715+ζ3717+ζ3718

magma: CharacterTable(G);