Properties

Label 33T45
Order \(47520\)
n \(33\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $33$
Transitive number $t$ :  $45$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (2,3)(4,11,17,25)(5,10,18,27)(6,12,16,26)(7,9)(13,22,31,30)(14,24,32,29)(15,23,33,28)(20,21), (1,12,23,4,30,21)(2,10,24,5,28,19)(3,11,22,6,29,20)(7,32,26)(8,33,27)(9,31,25)(13,17,15,16,14,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
7920:  $M_{11}$
15840:  22T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 11: $M_{11}$

Low degree siblings

36T16207

Siblings are shown 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 7, 9)(10,12)(13,15)(16,18)(20,21)(22,23)(26,27)(28,30)(31,33)$
$ 33 $ $1440$ $33$ $( 1,20, 7,24,18,27,11, 5,31,29,15, 3,19, 9,23,17,26,10, 4,33,28,14, 2,21, 8, 22,16,25,12, 6,32,30,13)$
$ 11, 11, 11 $ $720$ $11$ $( 1,19, 8,24,17,25,11, 4,32,29,14)( 2,20, 9,22,18,26,12, 5,33,30,15) ( 3,21, 7,23,16,27,10, 6,31,28,13)$
$ 22, 11 $ $2160$ $22$ $( 1,21, 8,23,17,27,11, 6,32,28,14, 3,19, 7,24,16,25,10, 4,31,29,13) ( 2,20, 9,22,18,26,12, 5,33,30,15)$
$ 33 $ $1440$ $33$ $( 1,15,28,32, 5,10,25,18,23, 8,20, 3,14,30,31, 4,12,27,17,22, 7,19, 2,13,29, 33, 6,11,26,16,24, 9,21)$
$ 11, 11, 11 $ $720$ $11$ $( 1,14,29,32, 4,11,25,17,24, 8,19)( 2,15,30,33, 5,12,26,18,22, 9,20) ( 3,13,28,31, 6,10,27,16,23, 7,21)$
$ 22, 11 $ $2160$ $22$ $( 1,13,29,31, 4,10,25,16,24, 7,19, 3,14,28,32, 6,11,27,17,23, 8,21) ( 2,15,30,33, 5,12,26,18,22, 9,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $165$ $2$ $( 4,25)( 5,26)( 6,27)( 7,31)( 8,32)( 9,33)(10,21)(11,19)(12,20)(13,23)(14,24) (15,22)$
$ 6, 6, 6, 6, 3, 3, 3 $ $330$ $6$ $( 1, 2, 3)( 4,26, 6,25, 5,27)( 7,32, 9,31, 8,33)(10,19,12,21,11,20) (13,24,15,23,14,22)(16,17,18)(28,29,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $495$ $2$ $( 2, 3)( 4,25)( 5,27)( 6,26)( 7,33)( 8,32)( 9,31)(10,20)(11,19)(12,21)(13,22) (14,24)(15,23)(16,18)(28,30)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $990$ $4$ $( 4,24,25,14)( 5,22,26,15)( 6,23,27,13)( 7,21,31,10)( 8,19,32,11)( 9,20,33,12)$
$ 12, 12, 3, 3, 3 $ $1980$ $12$ $( 1, 2, 3)( 4,22,27,14, 5,23,25,15, 6,24,26,13)( 7,19,33,10, 8,20,31,11, 9,21, 32,12)(16,17,18)(28,29,30)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $2970$ $4$ $( 2, 3)( 4,24,25,14)( 5,23,26,13)( 6,22,27,15)( 7,20,31,12)( 8,19,32,11) ( 9,21,33,10)(16,18)(28,30)$
$ 24, 6, 3 $ $1980$ $24$ $( 1,28, 2,29, 3,30)( 4,31,15,19,27, 9,24,10, 5,32,13,20,25, 7,22,11, 6,33,14, 21,26, 8,23,12)(16,18,17)$
$ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ $990$ $8$ $( 1,29)( 2,30)( 3,28)( 4,32,14,19,25, 8,24,11)( 5,33,15,20,26, 9,22,12) ( 6,31,13,21,27, 7,23,10)$
$ 8, 8, 8, 2, 2, 2, 2, 1 $ $2970$ $8$ $( 1,30)( 2,29)( 3,28)( 4,33,14,20,25, 9,24,12)( 5,32,15,19,26, 8,22,11) ( 6,31,13,21,27, 7,23,10)(17,18)$
$ 24, 6, 3 $ $1980$ $24$ $( 1,28, 2,29, 3,30)( 4,10,22, 8,27,20,14,31, 5,11,23, 9,25,21,15,32, 6,12,24, 7,26,19,13,33)(16,18,17)$
$ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ $990$ $8$ $( 1,29)( 2,30)( 3,28)( 4,11,24, 8,25,19,14,32)( 5,12,22, 9,26,20,15,33) ( 6,10,23, 7,27,21,13,31)$
$ 8, 8, 8, 2, 2, 2, 2, 1 $ $2970$ $8$ $( 1,30)( 2,29)( 3,28)( 4,12,24, 9,25,20,14,33)( 5,11,22, 8,26,19,15,32) ( 6,10,23, 7,27,21,13,31)(17,18)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $1584$ $5$ $( 1, 8,17,14,11)( 2, 9,18,15,12)( 3, 7,16,13,10)( 4,19,24,32,29) ( 5,20,22,33,30)( 6,21,23,31,28)$
$ 15, 15, 3 $ $3168$ $15$ $( 1, 9,16,14,12, 3, 8,18,13,11, 2, 7,17,15,10)( 4,20,23,32,30, 6,19,22,31,29, 5,21,24,33,28)(25,26,27)$
$ 10, 10, 5, 5, 2, 1 $ $4752$ $10$ $( 1, 8,17,14,11)( 2, 7,18,13,12, 3, 9,16,15,10)( 4,19,24,32,29) ( 5,21,22,31,30, 6,20,23,33,28)(26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $880$ $3$ $( 1,28,18)( 2,29,16)( 3,30,17)( 4,13,12)( 5,14,10)( 6,15,11)( 7, 9, 8) (19,27,22)(20,25,23)(21,26,24)(31,33,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $440$ $3$ $( 1,29,17)( 2,30,18)( 3,28,16)( 4,14,11)( 5,15,12)( 6,13,10)(19,25,24) (20,26,22)(21,27,23)$
$ 6, 6, 6, 3, 3, 3, 2, 2, 1, 1 $ $1320$ $6$ $( 1,30,17, 2,29,18)( 3,28,16)( 4,15,11, 5,14,12)( 6,13,10)( 8, 9) (19,26,24,20,25,22)(21,27,23)(32,33)$
$ 6, 6, 6, 6, 3, 3, 3 $ $2640$ $6$ $( 1,20,23, 4,15,16)( 2,21,24, 5,13,17)( 3,19,22, 6,14,18)( 7,11,30)( 8,12,28) ( 9,10,29)(25,33,27,32,26,31)$
$ 6, 6, 6, 3, 3, 3, 2, 2, 2 $ $1320$ $6$ $( 1,19,24, 4,14,17)( 2,20,22, 5,15,18)( 3,21,23, 6,13,16)( 7,10,28)( 8,11,29) ( 9,12,30)(25,32)(26,33)(27,31)$
$ 6, 6, 6, 6, 3, 2, 2, 2 $ $3960$ $6$ $( 1,21,24, 6,14,16)( 2,20,22, 5,15,18)( 3,19,23, 4,13,17)( 7,11,28, 8,10,29) ( 9,12,30)(25,31)(26,33)(27,32)$

Group invariants

Order:  $47520=2^{5} \cdot 3^{3} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table: Data not available.