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

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.