# Properties

 Label 37T7 Degree $37$ Order $444$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $C_{37}:C_{12}$

Show commands: Magma

magma: G := TransitiveGroup(37, 7);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$

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

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/18$

## Group invariants

 Order: $444=2^{2} \cdot 3 \cdot 37$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 444.7 magma: IdentifyGroup(G); Character table:

 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 37A1 37A2 37A3 Size 1 37 37 37 37 37 37 37 37 37 37 37 12 12 12 2 P 1A 1A 3A-1 3A1 2A 2A 3A1 3A-1 6A1 6A-1 6A-1 6A1 37A3 37A1 37A2 3 P 1A 2A 1A 1A 4A-1 4A1 2A 2A 4A1 4A-1 4A1 4A-1 37A1 37A2 37A3 37 P 1A 2A 3A1 3A-1 4A1 4A-1 6A1 6A-1 12A1 12A-1 12A5 12A-5 1A 1A 1A Type 444.7.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 444.7.1b R $1$ $1$ $1$ $1$ $−1$ $−1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ 444.7.1c1 C $1$ $1$ $ζ3−1$ $ζ3$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $1$ $1$ $1$ 444.7.1c2 C $1$ $1$ $ζ3$ $ζ3−1$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $1$ $1$ $1$ 444.7.1d1 C $1$ $−1$ $1$ $1$ $−i$ $i$ $−1$ $−1$ $i$ $−i$ $i$ $−i$ $1$ $1$ $1$ 444.7.1d2 C $1$ $−1$ $1$ $1$ $i$ $−i$ $−1$ $−1$ $−i$ $i$ $−i$ $i$ $1$ $1$ $1$ 444.7.1e1 C $1$ $1$ $ζ3−1$ $ζ3$ $−1$ $−1$ $ζ3$ $ζ3−1$ $−ζ3−1$ $−ζ3$ $−ζ3$ $−ζ3−1$ $1$ $1$ $1$ 444.7.1e2 C $1$ $1$ $ζ3$ $ζ3−1$ $−1$ $−1$ $ζ3−1$ $ζ3$ $−ζ3$ $−ζ3−1$ $−ζ3−1$ $−ζ3$ $1$ $1$ $1$ 444.7.1f1 C $1$ $−1$ $−ζ122$ $ζ124$ $−ζ123$ $ζ123$ $−ζ124$ $ζ122$ $−ζ125$ $ζ12$ $−ζ12$ $ζ125$ $1$ $1$ $1$ 444.7.1f2 C $1$ $−1$ $ζ124$ $−ζ122$ $ζ123$ $−ζ123$ $ζ122$ $−ζ124$ $ζ12$ $−ζ125$ $ζ125$ $−ζ12$ $1$ $1$ $1$ 444.7.1f3 C $1$ $−1$ $−ζ122$ $ζ124$ $ζ123$ $−ζ123$ $−ζ124$ $ζ122$ $ζ125$ $−ζ12$ $ζ12$ $−ζ125$ $1$ $1$ $1$ 444.7.1f4 C $1$ $−1$ $ζ124$ $−ζ122$ $−ζ123$ $ζ123$ $ζ122$ $−ζ124$ $−ζ12$ $ζ125$ $−ζ125$ $ζ12$ $1$ $1$ $1$ 444.7.12a1 R $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ37−17+ζ37−16+ζ37−15+ζ37−12+ζ37−9+ζ37−2+ζ372+ζ379+ζ3712+ζ3715+ζ3716+ζ3717$ $ζ37−18+ζ37−13+ζ37−7+ζ37−5+ζ37−4+ζ37−3+ζ373+ζ374+ζ375+ζ377+ζ3713+ζ3718$ $ζ37−14+ζ37−11+ζ37−10+ζ37−8+ζ37−6+ζ37−1+ζ37+ζ376+ζ378+ζ3710+ζ3711+ζ3714$ 444.7.12a2 R $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ37−14+ζ37−11+ζ37−10+ζ37−8+ζ37−6+ζ37−1+ζ37+ζ376+ζ378+ζ3710+ζ3711+ζ3714$ $ζ37−17+ζ37−16+ζ37−15+ζ37−12+ζ37−9+ζ37−2+ζ372+ζ379+ζ3712+ζ3715+ζ3716+ζ3717$ $ζ37−18+ζ37−13+ζ37−7+ζ37−5+ζ37−4+ζ37−3+ζ373+ζ374+ζ375+ζ377+ζ3713+ζ3718$ 444.7.12a3 R $12$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $ζ37−18+ζ37−13+ζ37−7+ζ37−5+ζ37−4+ζ37−3+ζ373+ζ374+ζ375+ζ377+ζ3713+ζ3718$ $ζ37−14+ζ37−11+ζ37−10+ζ37−8+ζ37−6+ζ37−1+ζ37+ζ376+ζ378+ζ3710+ζ3711+ζ3714$ $ζ37−17+ζ37−16+ζ37−15+ζ37−12+ζ37−9+ζ37−2+ζ372+ζ379+ζ3712+ζ3715+ζ3716+ζ3717$

magma: CharacterTable(G);