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

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.