Properties

Label 37T9
Order \(1332\)
n \(37\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_{37}$

Related objects

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Group action invariants

Degree $n$ :  $37$
Transitive number $t$ :  $9$
Group :  $F_{37}$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

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

Group invariants

Order:  $1332=2^{2} \cdot 3^{2} \cdot 37$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1332, 31]
Character table: Data not available.