# Properties

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

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

 Label Cycle Type Size Order Index Representative 1A $1^{37}$ $1$ $1$ $0$ $()$ 2A $2^{18},1$ $37$ $2$ $18$ $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ 3A1 $3^{12},1$ $37$ $3$ $24$ $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17)(12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$ 3A-1 $3^{12},1$ $37$ $3$ $24$ $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13)(12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$ 6A1 $6^{6},1$ $37$ $6$ $30$ $( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19)( 7,15, 9,32,24,30)(10,22,13,29,17,26)$ 6A-1 $6^{6},1$ $37$ $6$ $30$ $( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25)( 7,30,24,32, 9,15)(10,26,17,29,13,22)$ 9A1 $9^{4},1$ $37$ $9$ $32$ $( 2, 8,13,11,34,10,27,35,17)( 3,15,25,21,30,19,16,32,33)( 4,22,37,31,26,28, 5,29,12)( 6,36,24,14,18, 9,20,23, 7)$ 9A-1 $9^{4},1$ $37$ $9$ $32$ $( 2,13,34,27,17, 8,11,10,35)( 3,25,30,16,33,15,21,19,32)( 4,37,26, 5,12,22,31,28,29)( 6,24,18,20, 7,36,14, 9,23)$ 9A2 $9^{4},1$ $37$ $9$ $32$ $( 2,35,10,11, 8,17,27,34,13)( 3,32,19,21,15,33,16,30,25)( 4,29,28,31,22,12, 5,26,37)( 6,23, 9,14,36, 7,20,18,24)$ 9A-2 $9^{4},1$ $37$ $9$ $32$ $( 2,17,35,27,10,34,11,13, 8)( 3,33,32,16,19,30,21,25,15)( 4,12,29, 5,28,26,31,37,22)( 6, 7,23,20, 9,18,14,24,36)$ 9A4 $9^{4},1$ $37$ $9$ $32$ $( 2,10, 8,27,13,35,11,17,34)( 3,19,15,16,25,32,21,33,30)( 4,28,22, 5,37,29,31,12,26)( 6, 9,36,20,24,23,14, 7,18)$ 9A-4 $9^{4},1$ $37$ $9$ $32$ $( 2,34,17,11,35,13,27, 8,10)( 3,30,33,21,32,25,16,15,19)( 4,26,12,31,29,37, 5,22,28)( 6,18, 7,14,23,24,20,36, 9)$ 18A1 $18^{2},1$ $37$ $18$ $34$ $( 2,22,35,12,10, 5,11,26, 8,37,17, 4,27,29,34,28,13,31)( 3, 6,32,23,19, 9,21,14,15,36,33, 7,16,20,30,18,25,24)$ 18A-1 $18^{2},1$ $37$ $18$ $34$ $( 2,29, 8,12,13, 4,11,22,34,37,10,31,27,26,35,28,17, 5)( 3,20,15,23,25, 7,21, 6,30,36,19,24,16,14,32,18,33, 9)$ 18A5 $18^{2},1$ $37$ $18$ $34$ $( 2,31,13,28,34,29,27, 4,17,37, 8,26,11, 5,10,12,35,22)( 3,24,25,18,30,20,16, 7,33,36,15,14,21, 9,19,23,32, 6)$ 18A-5 $18^{2},1$ $37$ $18$ $34$ $( 2,26,34,12,17,31,11,29,35,37,13, 5,27,22, 8,28,10, 4)( 3,14,30,23,33,24,21,20,32,36,25, 9,16, 6,15,18,19, 7)$ 18A7 $18^{2},1$ $37$ $18$ $34$ $( 2, 5,17,28,35,26,27,31,10,37,34,22,11, 4,13,12, 8,29)( 3, 9,33,18,32,14,16,24,19,36,30, 6,21, 7,25,23,15,20)$ 18A-7 $18^{2},1$ $37$ $18$ $34$ $( 2, 4,10,28, 8,22,27, 5,13,37,35,29,11,31,17,12,34,26)( 3, 7,19,18,15, 6,16, 9,25,36,32,20,21,24,33,23,30,14)$ 37A1 $37$ $18$ $37$ $36$ $( 1,29,20,11, 2,30,21,12, 3,31,22,13, 4,32,23,14, 5,33,24,15, 6,34,25,16, 7,35,26,17, 8,36,27,18, 9,37,28,19,10)$ 37A2 $37$ $18$ $37$ $36$ $( 1,20, 2,21, 3,22, 4,23, 5,24, 6,25, 7,26, 8,27, 9,28,10,29,11,30,12,31,13,32,14,33,15,34,16,35,17,36,18,37,19)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/18$

## 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);