# Properties

 Label 33T10 Degree $33$ Order $363$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{11}^2:C_3$

## Group action invariants

 Degree $n$: $33$ Transitive number $t$: $10$ Group: $C_{11}^2:C_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $11$ Generators: (1,23,12)(2,28,16)(3,33,20)(4,27,13)(5,32,17)(6,26,21)(7,31,14)(8,25,18)(9,30,22)(10,24,15)(11,29,19), (1,29,16)(2,23,20)(3,28,13)(4,33,17)(5,27,21)(6,32,14)(7,26,18)(8,31,22)(9,25,15)(10,30,19)(11,24,12)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 11: None

## Low degree siblings

33T10 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,13,14,15,16,17,18,19,20,21,22)(23,30,26,33,29,25,32,28,24,31,27)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,14,16,18,20,22,13,15,17,19,21)(23,26,29,32,24,27,30,33,25,28,31)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,15,18,21,13,16,19,22,14,17,20)(23,33,32,31,30,29,28,27,26,25,24)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,16,20,13,17,21,14,18,22,15,19)(23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,17,22,16,21,15,20,14,19,13,18)(23,25,27,29,31,33,24,26,28,30,32)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,18,13,19,14,20,15,21,16,22,17)(23,32,30,28,26,24,33,31,29,27,25)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,19,15,22,18,14,21,17,13,20,16)(23,28,33,27,32,26,31,25,30,24,29)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,20,17,14,22,19,16,13,21,18,15)(23,24,25,26,27,28,29,30,31,32,33)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,21,19,17,15,13,22,20,18,16,14)(23,31,28,25,33,30,27,24,32,29,26)$ $11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $11$ $(12,22,21,20,19,18,17,16,15,14,13)(23,27,31,24,28,32,25,29,33,26,30)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,13,14,15,16,17,18,19,20,21,22) (23,25,27,29,31,33,24,26,28,30,32)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,14,16,18,20,22,13,15,17,19,21) (23,32,30,28,26,24,33,31,29,27,25)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,15,18,21,13,16,19,22,14,17,20) (23,28,33,27,32,26,31,25,30,24,29)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,17,22,16,21,15,20,14,19,13,18) (23,31,28,25,33,30,27,24,32,29,26)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,18,13,19,14,20,15,21,16,22,17) (23,27,31,24,28,32,25,29,33,26,30)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,20,17,14,22,19,16,13,21,18,15) (23,30,26,33,29,25,32,28,24,31,27)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,21,19,17,15,13,22,20,18,16,14) (23,26,29,32,24,27,30,33,25,28,31)$ $11, 11, 11$ $3$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,21,20,19,18,17,16,15,14,13) (23,33,32,31,30,29,28,27,26,25,24)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,13,14,15,16,17,18,19,20,21,22) (23,31,28,25,33,30,27,24,32,29,26)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,14,16,18,20,22,13,15,17,19,21) (23,27,31,24,28,32,25,29,33,26,30)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,17,22,16,21,15,20,14,19,13,18) (23,26,29,32,24,27,30,33,25,28,31)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,18,13,19,14,20,15,21,16,22,17) (23,33,32,31,30,29,28,27,26,25,24)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,19,15,22,18,14,21,17,13,20,16) (23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 11$ $3$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,21,19,17,15,13,22,20,18,16,14) (23,32,30,28,26,24,33,31,29,27,25)$ $11, 11, 11$ $3$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,13,14,15,16,17,18,19,20,21,22) (23,26,29,32,24,27,30,33,25,28,31)$ $11, 11, 11$ $3$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,15,18,21,13,16,19,22,14,17,20) (23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 11$ $3$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,17,22,16,21,15,20,14,19,13,18) (23,32,30,28,26,24,33,31,29,27,25)$ $11, 11, 11$ $3$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,19,15,22,18,14,21,17,13,20,16) (23,24,25,26,27,28,29,30,31,32,33)$ $11, 11, 11$ $3$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,15,18,21,13,16,19,22,14,17,20) (23,24,25,26,27,28,29,30,31,32,33)$ $11, 11, 11$ $3$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,19,15,22,18,14,21,17,13,20,16) (23,30,26,33,29,25,32,28,24,31,27)$ $11, 11, 11$ $3$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,22,21,20,19,18,17,16,15,14,13) (23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 11$ $3$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,15,18,21,13,16,19,22,14,17,20) (23,30,26,33,29,25,32,28,24,31,27)$ $11, 11, 11$ $3$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,18,13,19,14,20,15,21,16,22,17) (23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 11$ $3$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,19,15,22,18,14,21,17,13,20,16) (23,25,27,29,31,33,24,26,28,30,32)$ $11, 11, 11$ $3$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,22,21,20,19,18,17,16,15,14,13) (23,24,25,26,27,28,29,30,31,32,33)$ $11, 11, 11$ $3$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,14,16,18,20,22,13,15,17,19,21) (23,29,24,30,25,31,26,32,27,33,28)$ $11, 11, 11$ $3$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,15,18,21,13,16,19,22,14,17,20) (23,25,27,29,31,33,24,26,28,30,32)$ $11, 11, 11$ $3$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,18,13,19,14,20,15,21,16,22,17) (23,24,25,26,27,28,29,30,31,32,33)$ $11, 11, 11$ $3$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,22,21,20,19,18,17,16,15,14,13) (23,30,26,33,29,25,32,28,24,31,27)$ $11, 11, 11$ $3$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,18,13,19,14,20,15,21,16,22,17) (23,30,26,33,29,25,32,28,24,31,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $121$ $3$ $( 1,12,23)( 2,16,28)( 3,20,33)( 4,13,27)( 5,17,32)( 6,21,26)( 7,14,31) ( 8,18,25)( 9,22,30)(10,15,24)(11,19,29)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $121$ $3$ $( 1,23,12)( 2,28,16)( 3,33,20)( 4,27,13)( 5,32,17)( 6,26,21)( 7,31,14) ( 8,25,18)( 9,30,22)(10,24,15)(11,29,19)$

## Group invariants

 Order: $363=3 \cdot 11^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [363, 2]
 Character table: not available.