Properties

Label 38T33
Order \(25308\)
n \(38\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $\PSL(2,37)$

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Group action invariants

Degree $n$ :  $38$
Transitive number $t$ :  $33$
Group :  $\PSL(2,37)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

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

Group invariants

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