Properties

Label 33T24
Order \(5346\)
n \(33\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
11:  $C_{11}$
22:  22T1

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 11: $C_{11}$

Low degree siblings

33T24 x 10

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 3, 2)(10,12,11)(13,15,14)(16,17,18)(22,23,24)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(28,30,29)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(13,14,15)(16,17,18)(22,23,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,24,23)(25,26,27) (28,29,30)(31,32,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(25,26,27) (28,30,29)(31,32,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 9, 8)(10,12,11)(16,18,17)(19,20,21)(25,26,27) (28,29,30)(31,32,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 2, 3)( 4, 6, 5)(10,11,12)(16,17,18)(19,20,21)(22,24,23)(25,26,27) (28,30,29)(31,32,33)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 3, 2)(10,11,12)(13,15,14)(19,20,21)(22,24,23)(25,26,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(13,15,14)(16,18,17)(19,20,21)(22,23,24) (25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 4, 6, 5)( 7, 9, 8)(19,20,21)(22,24,23)(25,26,27)(28,29,30)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $3$ $( 7, 9, 8)(13,14,15)(16,17,18)(22,24,23)(28,29,30)(31,33,32)$
$ 11, 11, 11 $ $243$ $11$ $( 1,22,11,31,21, 7,29,18, 5,25,13)( 2,23,12,32,19, 8,30,16, 6,26,14) ( 3,24,10,33,20, 9,28,17, 4,27,15)$
$ 11, 11, 11 $ $243$ $11$ $( 1,11,21,29, 5,13,22,31, 7,18,25)( 2,12,19,30, 6,14,23,32, 8,16,26) ( 3,10,20,28, 4,15,24,33, 9,17,27)$
$ 11, 11, 11 $ $243$ $11$ $( 1,21, 5,22, 7,25,11,29,13,31,18)( 2,19, 6,23, 8,26,12,30,14,32,16) ( 3,20, 4,24, 9,27,10,28,15,33,17)$
$ 11, 11, 11 $ $243$ $11$ $( 1, 5, 7,11,13,18,21,22,25,29,31)( 2, 6, 8,12,14,16,19,23,26,30,32) ( 3, 4, 9,10,15,17,20,24,27,28,33)$
$ 11, 11, 11 $ $243$ $11$ $( 1, 7,13,21,25,31, 5,11,18,22,29)( 2, 8,14,19,26,32, 6,12,16,23,30) ( 3, 9,15,20,27,33, 4,10,17,24,28)$
$ 11, 11, 11 $ $243$ $11$ $( 1,13,25, 5,18,29, 7,21,31,11,22)( 2,14,26, 6,16,30, 8,19,32,12,23) ( 3,15,27, 4,17,28, 9,20,33,10,24)$
$ 11, 11, 11 $ $243$ $11$ $( 1,25,18, 7,31,22,13, 5,29,21,11)( 2,26,16, 8,32,23,14, 6,30,19,12) ( 3,27,17, 9,33,24,15, 4,28,20,10)$
$ 11, 11, 11 $ $243$ $11$ $( 1,18,31,13,29,11,25, 7,22, 5,21)( 2,16,32,14,30,12,26, 8,23, 6,19) ( 3,17,33,15,28,10,27, 9,24, 4,20)$
$ 11, 11, 11 $ $243$ $11$ $( 1,31,29,25,22,21,18,13,11, 7, 5)( 2,32,30,26,23,19,16,14,12, 8, 6) ( 3,33,28,27,24,20,17,15,10, 9, 4)$
$ 11, 11, 11 $ $243$ $11$ $( 1,29,22,18,11, 5,31,25,21,13, 7)( 2,30,23,16,12, 6,32,26,19,14, 8) ( 3,28,24,17,10, 4,33,27,20,15, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $243$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(19,20)(23,24)(25,27)(28,29)(31,32)$
$ 22, 11 $ $243$ $22$ $( 1,22,11,32,21, 7,29,17, 5,26,14, 3,23,10,33,20, 8,28,18, 4,27,13) ( 2,24,12,31,19, 9,30,16, 6,25,15)$
$ 22, 11 $ $243$ $22$ $( 1,11,19,28, 5,14,24,32, 7,18,27)( 2,10,20,30, 6,13,22,31, 8,17,25, 3,12,21, 29, 4,15,23,33, 9,16,26)$
$ 22, 11 $ $243$ $22$ $( 1,21, 5,23, 9,26,12,29,15,32,18, 3,19, 4,24, 8,27,11,30,14,33,17) ( 2,20, 6,22, 7,25,10,28,13,31,16)$
$ 22, 11 $ $243$ $22$ $( 1, 5, 8,10,15,16,19,24,26,30,32)( 2, 4, 9,12,13,18,20,23,27,29,33, 3, 6, 7, 11,14,17,21,22,25,28,31)$
$ 22, 11 $ $243$ $22$ $( 1, 7,13,21,25,33, 4,10,17,22,29, 3, 8,15,19,27,31, 6,11,16,23,28) ( 2, 9,14,20,26,32, 5,12,18,24,30)$
$ 22, 11 $ $243$ $22$ $( 1,13,25, 4,17,29, 9,19,33,10,24, 2,15,26, 6,18,28, 7,21,31,12,22) ( 3,14,27, 5,16,30, 8,20,32,11,23)$
$ 22, 11 $ $243$ $22$ $( 1,25,17, 7,31,23,15, 6,29,20,12)( 2,27,18, 9,32,22,13, 5,30,19,10, 3,26,16, 8,33,24,14, 4,28,21,11)$
$ 22, 11 $ $243$ $22$ $( 1,18,33,15,30,12,25, 9,23, 4,20, 2,17,31,14,28,11,26, 8,24, 6,21) ( 3,16,32,13,29,10,27, 7,22, 5,19)$
$ 22, 11 $ $243$ $22$ $( 1,31,30,26,23,20,16,14,10, 9, 6)( 2,33,28,25,24,19,17,13,11, 8, 4, 3,32,29, 27,22,21,18,15,12, 7, 5)$
$ 22, 11 $ $243$ $22$ $( 1,29,24,16,12, 5,32,25,20,14, 9, 2,28,22,18,10, 4,33,27,21,13, 7) ( 3,30,23,17,11, 6,31,26,19,15, 8)$

Group invariants

Order:  $5346=2 \cdot 3^{5} \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.