# Properties

 Label 38T33 Degree $38$ Order $25308$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,37)$

## Group action invariants

 Degree $n$: $38$ Transitive number $t$: $33$ Group: $\PSL(2,37)$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (3,20,23,11,9,13,22,14,16,24,34,12,33,15,6,31,37,36)(4,10,35,26,8,29,7,17,25,27,19,28,32,30,18,21,38,5), (1,21,2)(3,7,20)(5,13,11)(6,30,37)(8,9,36)(10,26,18)(12,14,17)(15,35,34)(16,29,33)(19,22,24)(23,31,25)(27,28,38)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 19: 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1$ $1406$ $3$ $( 1,37, 4)( 2,21,22)( 3,20,16)( 5,13,35)( 6,11, 9)( 7,32,28)( 8,26,27) (10,12,18)(14,25,15)(17,31,24)(23,34,33)(30,36,38)$ $9, 9, 9, 9, 1, 1$ $1406$ $9$ $( 1,27,16,37, 8, 3, 4,26,20)( 2,11,32,21, 9,28,22, 6, 7)( 5,18,36,13,10,38,35, 12,30)(14,24,34,25,17,33,15,31,23)$ $9, 9, 9, 9, 1, 1$ $1406$ $9$ $( 1,16, 8, 4,20,27,37, 3,26)( 2,32, 9,22, 7,11,21,28, 6)( 5,36,10,35,30,18,13, 38,12)(14,34,17,15,23,24,25,33,31)$ $9, 9, 9, 9, 1, 1$ $1406$ $9$ $( 1, 8,20,37,26,16, 4,27, 3)( 2, 9, 7,21, 6,32,22,11,28)( 5,10,30,13,12,36,35, 18,38)(14,17,23,25,31,34,15,24,33)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1$ $703$ $2$ $( 1, 6)( 2,16)( 3,21)( 4, 9)( 5,24)( 7,27)( 8,32)(10,33)(11,37)(12,23)(13,17) (14,30)(15,38)(18,34)(20,22)(25,36)(26,28)(31,35)$ $6, 6, 6, 6, 6, 6, 1, 1$ $1406$ $6$ $( 1,11, 4, 6,37, 9)( 2, 3,22,16,21,20)( 5,17,35,24,13,31)( 7, 8,28,27,32,26) (10,23,18,33,12,34)(14,36,15,30,25,38)$ $18, 18, 1, 1$ $1406$ $18$ $( 1,32,20,11,26, 2, 4, 7, 3, 6, 8,22,37,28,16, 9,27,21)( 5,33,30,17,12,25,35, 34,38,24,10,14,13,23,36,31,18,15)$ $18, 18, 1, 1$ $1406$ $18$ $( 1,22,26, 9, 3,32,37, 2,27, 6,20,28, 4,21, 8,11,16, 7)( 5,14,12,31,38,33,13, 25,18,24,30,23,35,15,10,17,36,34)$ $18, 18, 1, 1$ $1406$ $18$ $( 1,28, 3,11,27,22, 4,32,16, 6,26,21,37, 7,20, 9, 8, 2)( 5,23,38,17,18,14,35, 33,36,24,12,15,13,34,30,31,10,25)$ $37, 1$ $684$ $37$ $( 1,32,17,18,30,31,16, 9,13,29,25,36,20,26, 7,15, 6,34,11,38,21, 8, 5, 2,27, 10,37,14, 4,33, 3,22,28,12,23,19,35)$ $37, 1$ $684$ $37$ $( 1,16,20,11,27, 3,35,31,36,34, 2,33,19,30,25, 6, 5, 4,23,18,29,15, 8,14,12, 17,13, 7,21,37,28,32, 9,26,38,10,22)$ $19, 19$ $1332$ $19$ $( 1,34, 7, 9,32, 6,13,27,24,18,38,36,37,21,26,25,14,17,10)( 2,16,23,35,20,22, 33,28,19,12,15, 4, 3, 8,30,31,29,11, 5)$ $19, 19$ $1332$ $19$ $( 1,36, 9,26,13,17,18,34,37,32,25,27,10,38, 7,21, 6,14,24)( 2, 4,35,30,33,11, 12,16, 3,20,31,28, 5,15,23, 8,22,29,19)$ $19, 19$ $1332$ $19$ $( 1,27,26, 7,18,14,32,36,10,13,21,34,24,25, 9,38,17, 6,37)( 2,28,30,23,12,29, 20, 4, 5,33, 8,16,19,31,35,15,11,22, 3)$ $19, 19$ $1332$ $19$ $( 1, 6,38,25,34,13,36,14, 7,27,37,17, 9,24,21,10,32,18,26)( 2,22,15,31,16,33, 4,29,23,28, 3,11,35,19, 8, 5,20,12,30)$ $19, 19$ $1332$ $19$ $( 1,17,25,21,36,18,27, 6, 9,34,10,14,26,37,38,24,13,32, 7)( 2,11,31, 8, 4,12, 28,22,35,16, 5,29,30, 3,15,19,33,20,23)$ $19, 19$ $1332$ $19$ $( 1,14,21,38,27,32,34,17,26,36,24, 6, 7,10,25,37,18,13, 9)( 2,29, 8,15,28,20, 16,11,30, 4,19,22,23, 5,31, 3,12,33,35)$ $19, 19$ $1332$ $19$ $( 1,13,37,10, 6,36,17,32,38,14, 9,18,25, 7,24,26,34,27,21)( 2,33, 3, 5,22, 4, 11,20,15,29,35,12,31,23,19,30,16,28, 8)$ $19, 19$ $1332$ $19$ $( 1,18,10,24,17,27,14,13,25, 6,26,32,21, 9,37, 7,36,34,38)( 2,12, 5,19,11,28, 29,33,31,22,30,20, 8,35, 3,23, 4,16,15)$ $19, 19$ $1332$ $19$ $( 1,32,24,37,14,34, 6,18,21,17, 7,13,38,26,10, 9,27,36,25)( 2,20,19, 3,29,16, 22,12, 8,11,23,33,15,30, 5,35,28, 4,31)$

## Group invariants

 Order: $25308=2^{2} \cdot 3^{2} \cdot 19 \cdot 37$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table: not available.