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

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.