Properties

Label 33T45
Degree $33$
Order $47520$
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)
$|\Aut(F/K)|$:  $1$
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)

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

Group invariants

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