# Properties

 Label 37T9 Degree $37$ Order $1332$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{37}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(37, 9);

## Group action invariants

 Degree $n$: $37$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $9$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_{37}$ 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: (1,2,4,8,16,32,27,17,34,31,25,13,26,15,30,23,9,18,36,35,33,29,21,5,10,20,3,6,12,24,11,22,7,14,28,19), (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)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$9$:  $C_9$
$12$:  $C_{12}$
$18$:  $C_{18}$
$36$:  $C_{36}$

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 $4^{9},1$ $37$ $4$ $27$ $( 1,31,36, 6)( 2,25,35,12)( 3,19,34,18)( 4,13,33,24)( 5, 7,32,30)( 8,26,29,11)( 9,20,28,17)(10,14,27,23)(15,21,22,16)$ 4A-1 $4^{9},1$ $37$ $4$ $27$ $( 1, 6,36,31)( 2,12,35,25)( 3,18,34,19)( 4,24,33,13)( 5,30,32, 7)( 8,11,29,26)( 9,17,28,20)(10,23,27,14)(15,16,22,21)$ 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 $6^{6},1$ $37$ $6$ $30$ $( 1,27,26,36,10,11)( 2,17,15,35,20,22)( 3, 7, 4,34,30,33)( 5,24,19,32,13,18)( 6,14, 8,31,23,29)( 9,21,12,28,16,25)$ 9A1 $9^{4},1$ $37$ $9$ $32$ $( 1, 7,12,10,33, 9,26,34,16)( 2,14,24,20,29,18,15,31,32)( 3,21,36,30,25,27, 4,28,11)( 5,35,23,13,17, 8,19,22, 6)$ 9A-1 $9^{4},1$ $37$ $9$ $32$ $( 1,16,34,26, 9,33,10,12, 7)( 2,32,31,15,18,29,20,24,14)( 3,11,28, 4,27,25,30,36,21)( 5, 6,22,19, 8,17,13,23,35)$ 9A2 $9^{4},1$ $37$ $9$ $32$ $( 1,12,33,26,16, 7,10, 9,34)( 2,24,29,15,32,14,20,18,31)( 3,36,25, 4,11,21,30,27,28)( 5,23,17,19, 6,35,13, 8,22)$ 9A-2 $9^{4},1$ $37$ $9$ $32$ $( 1,34, 9,10, 7,16,26,33,12)( 2,31,18,20,14,32,15,29,24)( 3,28,27,30,21,11, 4,25,36)( 5,22, 8,13,35, 6,19,17,23)$ 9A4 $9^{4},1$ $37$ $9$ $32$ $( 1,33,16,10,34,12,26, 7, 9)( 2,29,32,20,31,24,15,14,18)( 3,25,11,30,28,36, 4,21,27)( 5,17, 6,13,22,23,19,35, 8)$ 9A-4 $9^{4},1$ $37$ $9$ $32$ $( 1, 9, 7,26,12,34,10,16,33)( 2,18,14,15,24,31,20,32,29)( 3,27,21, 4,36,28,30,11,25)( 5, 8,35,19,23,22,13, 6,17)$ 12A1 $12^{3},1$ $37$ $12$ $33$ $( 1,23,11,31,10, 8,36,14,26, 6,27,29)( 2, 9,22,25,20,16,35,28,15,12,17,21)( 3,32,33,19,30,24,34, 5, 4,18, 7,13)$ 12A-1 $12^{3},1$ $37$ $12$ $33$ $( 1,14,11, 6,10,29,36,23,26,31,27, 8)( 2,28,22,12,20,21,35, 9,15,25,17,16)( 3, 5,33,18,30,13,34,32, 4,19, 7,24)$ 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 $12^{3},1$ $37$ $12$ $33$ $( 1, 8,27,31,26,23,36,29,10, 6,11,14)( 2,16,17,25,15, 9,35,21,20,12,22,28)( 3,24, 7,19, 4,32,34,13,30,18,33, 5)$ 18A1 $18^{2},1$ $37$ $18$ $34$ $( 1,25,33,11,16,30,10,28,34,36,12, 4,26,21, 7,27, 9, 3)( 2,13,29,22,32,23,20,19,31,35,24, 8,15, 5,14,17,18, 6)$ 18A-1 $18^{2},1$ $37$ $18$ $34$ $( 1,30,12,27,33,28,26, 3,16,36, 7,25,10, 4, 9,11,34,21)( 2,23,24,17,29,19,15, 6,32,35,14,13,20, 8,18,22,31, 5)$ 18A5 $18^{2},1$ $37$ $18$ $34$ $( 1, 3, 9,27, 7,21,26, 4,12,36,34,28,10,30,16,11,33,25)( 2, 6,18,17,14, 5,15, 8,24,35,31,19,20,23,32,22,29,13)$ 18A-5 $18^{2},1$ $37$ $18$ $34$ $( 1,21,34,11, 9, 4,10,25, 7,36,16, 3,26,28,33,27,12,30)( 2, 5,31,22,18, 8,20,13,14,35,32, 6,15,19,29,17,24,23)$ 18A7 $18^{2},1$ $37$ $18$ $34$ $( 1, 4,16,27,34,25,26,30, 9,36,33,21,10, 3,12,11, 7,28)( 2, 8,32,17,31,13,15,23,18,35,29, 5,20, 6,24,22,14,19)$ 18A-7 $18^{2},1$ $37$ $18$ $34$ $( 1,28, 7,11,12, 3,10,21,33,36, 9,30,26,25,34,27,16, 4)( 2,19,14,22,24, 6,20, 5,29,35,18,23,15,13,31,17,32, 8)$ 36A1 $36,1$ $37$ $36$ $35$ $( 1,24,21,23,34, 2,11, 5, 9,31, 4,22,10,18,25, 8, 7,20,36,13,16,14, 3,35,26,32,28, 6,33,15,27,19,12,29,30,17)$ 36A-1 $36,1$ $37$ $36$ $35$ $( 1, 2, 4, 8,16,32,27,17,34,31,25,13,26,15,30,23, 9,18,36,35,33,29,21, 5,10,20, 3, 6,12,24,11,22, 7,14,28,19)$ 36A5 $36,1$ $37$ $36$ $35$ $( 1,13,21,14,34,35,11,32, 9, 6, 4,15,10,19,25,29, 7,17,36,24,16,23, 3, 2,26, 5,28,31,33,22,27,18,12, 8,30,20)$ 36A-5 $36,1$ $37$ $36$ $35$ $( 1,22, 3,29, 9,13,27, 2, 7, 6,21,18,26,17, 4,14,12, 5,36,15,34, 8,28,24,10,35,30,31,16,19,11,20,33,23,25,32)$ 36A7 $36,1$ $37$ $36$ $35$ $( 1,20,30, 8,12,18,27,22,33,31,28, 5,26, 2, 3,23,16,24,36,17, 7,29,25,19,10,15, 4, 6, 9,32,11,35,34,14,21,13)$ 36A-7 $36,1$ $37$ $36$ $35$ $( 1,19,28,14, 7,22,11,24,12, 6, 3,20,10, 5,21,29,33,35,36,18, 9,23,30,15,26,13,25,31,34,17,27,32,16, 8, 4, 2)$ 36A11 $36,1$ $37$ $36$ $35$ $( 1,18,28,23, 7,15,11,13,12,31, 3,17,10,32,21, 8,33, 2,36,19, 9,14,30,22,26,24,25, 6,34,20,27, 5,16,29, 4,35)$ 36A-11 $36,1$ $37$ $36$ $35$ $( 1,17,30,29,12,19,27,15,33, 6,28,32,26,35, 3,14,16,13,36,20, 7, 8,25,18,10,22, 4,31, 9, 5,11, 2,34,23,21,24)$ 36A13 $36,1$ $37$ $36$ $35$ $( 1,32,25,23,33,20,11,19,16,31,30,35,10,24,28, 8,34,15,36, 5,12,14, 4,17,26,18,21, 6, 7, 2,27,13, 9,29, 3,22)$ 36A-13 $36,1$ $37$ $36$ $35$ $( 1,35, 4,29,16, 5,27,20,34, 6,25,24,26,22,30,14, 9,19,36, 2,33, 8,21,32,10,17, 3,31,12,13,11,15, 7,23,28,18)$ 36A17 $36,1$ $37$ $36$ $35$ $( 1, 5,25,14,33,17,11,18,16, 6,30, 2,10,13,28,29,34,22,36,32,12,23, 4,20,26,19,21,31, 7,35,27,24, 9, 8, 3,15)$ 36A-17 $36,1$ $37$ $36$ $35$ $( 1,15, 3, 8, 9,24,27,35, 7,31,21,19,26,20, 4,23,12,32,36,22,34,29,28,13,10, 2,30, 6,16,18,11,17,33,14,25, 5)$ 37A $37$ $36$ $37$ $36$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/18$

## Group invariants

 Order: $1332=2^{2} \cdot 3^{2} \cdot 37$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 1332.31 magma: IdentifyGroup(G); Character table: 37 x 37 character table

magma: CharacterTable(G);