# Properties

 Label 37T4 Degree $37$ Order $148$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{37}:C_{4}$

Show commands: Magma

magma: G := TransitiveGroup(37, 4);

## Group action invariants

 Degree $n$: $37$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{37}:C_{4}$ 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: (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (1,31,36,6)(2,25,35,12)(3,19,34,18)(4,13,33,24)(5,7,32,30)(8,26,29,11)(9,20,28,17)(10,14,27,23)(15,21,22,16) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

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 $2^{18},1$ $37$ $2$ $18$ $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ 4A1 $4^{9},1$ $37$ $4$ $27$ $( 2,32,37, 7)( 3,26,36,13)( 4,20,35,19)( 5,14,34,25)( 6, 8,33,31)( 9,27,30,12)(10,21,29,18)(11,15,28,24)(16,22,23,17)$ 4A-1 $4^{9},1$ $37$ $4$ $27$ $( 2, 7,37,32)( 3,13,36,26)( 4,19,35,20)( 5,25,34,14)( 6,31,33, 8)( 9,12,30,27)(10,18,29,21)(11,24,28,15)(16,17,23,22)$ 37A1 $37$ $4$ $37$ $36$ $( 1,29,20,11, 2,30,21,12, 3,31,22,13, 4,32,23,14, 5,33,24,15, 6,34,25,16, 7,35,26,17, 8,36,27,18, 9,37,28,19,10)$ 37A2 $37$ $4$ $37$ $36$ $( 1,28,18, 8,35,25,15, 5,32,22,12, 2,29,19, 9,36,26,16, 6,33,23,13, 3,30,20,10,37,27,17, 7,34,24,14, 4,31,21,11)$ 37A3 $37$ $4$ $37$ $36$ $( 1,23, 8,30,15,37,22, 7,29,14,36,21, 6,28,13,35,20, 5,27,12,34,19, 4,26,11,33,18, 3,25,10,32,17, 2,24, 9,31,16)$ 37A4 $37$ $4$ $37$ $36$ $( 1,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ 37A5 $37$ $4$ $37$ $36$ $( 1,36,34,32,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,37,35,33,31,29,27,25,23,21,19,17,15,13,11, 9, 7, 5, 3)$ 37A8 $37$ $4$ $37$ $36$ $( 1,35,32,29,26,23,20,17,14,11, 8, 5, 2,36,33,30,27,24,21,18,15,12, 9, 6, 3,37,34,31,28,25,22,19,16,13,10, 7, 4)$ 37A9 $37$ $4$ $37$ $36$ $( 1,34,30,26,22,18,14,10, 6, 2,35,31,27,23,19,15,11, 7, 3,36,32,28,24,20,16,12, 8, 4,37,33,29,25,21,17,13, 9, 5)$ 37A10 $37$ $4$ $37$ $36$ $( 1,33,28,23,18,13, 8, 3,35,30,25,20,15,10, 5,37,32,27,22,17,12, 7, 2,34,29,24,19,14, 9, 4,36,31,26,21,16,11, 6)$ 37A15 $37$ $4$ $37$ $36$ $( 1,30,22,14, 6,35,27,19,11, 3,32,24,16, 8,37,29,21,13, 5,34,26,18,10, 2,31,23,15, 7,36,28,20,12, 4,33,25,17, 9)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/18$

## Group invariants

 Order: $148=2^{2} \cdot 37$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 148.3 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 37A1 37A2 37A3 37A4 37A5 37A8 37A9 37A10 37A15 Size 1 37 37 37 4 4 4 4 4 4 4 4 4 2 P 1A 1A 2A 2A 37A3 37A9 37A5 37A2 37A4 37A1 37A8 37A10 37A15 37 P 1A 2A 4A-1 4A1 37A10 37A5 37A8 37A3 37A1 37A9 37A2 37A15 37A4 Type 148.3.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 148.3.1b R $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 148.3.1c1 C $1$ $−1$ $−i$ $i$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 148.3.1c2 C $1$ $−1$ $i$ $−i$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 148.3.4a1 R $4$ $0$ $0$ $0$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−18+ζ37−3+ζ373+ζ3718$ 148.3.4a2 R $4$ $0$ $0$ $0$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−13+ζ37−4+ζ374+ζ3713$ 148.3.4a3 R $4$ $0$ $0$ $0$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−12+ζ37−2+ζ372+ζ3712$ 148.3.4a4 R $4$ $0$ $0$ $0$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−11+ζ37−8+ζ378+ζ3711$ 148.3.4a5 R $4$ $0$ $0$ $0$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−17+ζ37−9+ζ379+ζ3717$ 148.3.4a6 R $4$ $0$ $0$ $0$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ 148.3.4a7 R $4$ $0$ $0$ $0$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−7+ζ37−5+ζ375+ζ377$ 148.3.4a8 R $4$ $0$ $0$ $0$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−16+ζ37−15+ζ3715+ζ3716$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−6+ζ37−1+ζ37+ζ376$ 148.3.4a9 R $4$ $0$ $0$ $0$ $ζ37−6+ζ37−1+ζ37+ζ376$ $ζ37−12+ζ37−2+ζ372+ζ3712$ $ζ37−18+ζ37−3+ζ373+ζ3718$ $ζ37−13+ζ37−4+ζ374+ζ3713$ $ζ37−7+ζ37−5+ζ375+ζ377$ $ζ37−11+ζ37−8+ζ378+ζ3711$ $ζ37−17+ζ37−9+ζ379+ζ3717$ $ζ37−14+ζ37−10+ζ3710+ζ3714$ $ζ37−16+ζ37−15+ζ3715+ζ3716$

magma: CharacterTable(G);