# Properties

 Label 33T36 Degree $33$ Order $26730$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

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## Group action invariants

 Degree $n$: $33$ Transitive number $t$: $36$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,8,14,19,26,32,4,12,16,23,30)(2,7,15,21,27,31,5,11,17,22,28,3,9,13,20,25,33,6,10,18,24,29), (1,28,12,24,25,2,30,10,23,26)(3,29,11,22,27)(4,5)(7,15,32,19,16,9,13,31,20,18)(8,14,33,21,17)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$55$:  $C_{11}:C_5$
$110$:  22T5

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 11: $C_{11}:C_5$

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $110$ $3$ $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(16,17,18)(22,24,23)(28,30,29)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $110$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(22,23,24) (25,26,27)(28,30,29)$ $3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $22$ $3$ $( 7, 8, 9)(13,14,15)(16,17,18)(22,23,24)(25,26,27)(28,29,30)$ $11, 11, 11$ $1215$ $11$ $( 1,14,26, 4,16,30, 8,19,32,12,23)( 2,15,27, 5,17,28, 9,20,33,10,24) ( 3,13,25, 6,18,29, 7,21,31,11,22)$ $11, 11, 11$ $1215$ $11$ $( 1,26,16, 8,32,23,14, 4,30,19,12)( 2,27,17, 9,33,24,15, 5,28,20,10) ( 3,25,18, 7,31,22,13, 6,29,21,11)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $243$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(19,21)(23,24)(25,27)(28,29)(31,32)$ $22, 11$ $1215$ $22$ $( 1,14,27, 6,17,29, 9,19,31,12,22, 3,15,26, 4,16,30, 8,20,33,10,24) ( 2,13,25, 5,18,28, 7,21,32,11,23)$ $22, 11$ $1215$ $22$ $( 1,26,16, 8,33,24,14, 5,29,20,10, 2,25,17, 7,31,23,15, 4,30,19,11) ( 3,27,18, 9,32,22,13, 6,28,21,12)$ $5, 5, 5, 5, 5, 5, 1, 1, 1$ $891$ $5$ $( 4,30,14,11,17)( 5,28,15,12,18)( 6,29,13,10,16)( 7,23,26,20,31) ( 8,24,27,21,32)( 9,22,25,19,33)$ $15, 5, 5, 5, 3$ $1782$ $15$ $( 1, 3, 2)( 4,29,13,10,17, 5,30,14,11,18, 6,28,15,12,16)( 7,22,25,19,33) ( 8,23,26,20,31)( 9,24,27,21,32)$ $10, 10, 5, 5, 2, 1$ $2673$ $10$ $( 2, 3)( 4,30,15,11,18, 6,28,14,12,17)( 5,29,13,10,16)( 7,24,25,21,31) ( 8,23,26,20,32, 9,22,27,19,33)$ $5, 5, 5, 5, 5, 5, 1, 1, 1$ $891$ $5$ $( 4,14,17,30,11)( 5,15,18,28,12)( 6,13,16,29,10)( 7,26,31,23,20) ( 8,27,32,24,21)( 9,25,33,22,19)$ $15, 5, 5, 5, 3$ $1782$ $15$ $( 1, 3, 2)( 4,14,18,30,11, 5,15,16,28,12, 6,13,17,29,10)( 7,26,31,22,19) ( 8,27,32,23,20)( 9,25,33,24,21)$ $10, 10, 5, 5, 2, 1$ $2673$ $10$ $( 2, 3)( 4,15,17,30,12, 6,13,16,28,11)( 5,14,18,29,10)( 7,26,32,23,20) ( 8,25,33,22,21, 9,27,31,24,19)$ $5, 5, 5, 5, 5, 5, 1, 1, 1$ $891$ $5$ $( 4,17,11,14,30)( 5,18,12,15,28)( 6,16,10,13,29)( 7,31,20,26,23) ( 8,32,21,27,24)( 9,33,19,25,22)$ $15, 5, 5, 5, 3$ $1782$ $15$ $( 1, 3, 2)( 4,18,12,15,30, 5,16,10,13,28, 6,17,11,14,29)( 7,31,20,26,22) ( 8,32,21,27,23)( 9,33,19,25,24)$ $10, 10, 5, 5, 2, 1$ $2673$ $10$ $( 2, 3)( 4,18,11,15,29, 5,17,12,14,30)( 6,16,10,13,28)( 7,32,19,27,23) ( 8,31,20,26,24, 9,33,21,25,22)$ $5, 5, 5, 5, 5, 5, 1, 1, 1$ $891$ $5$ $( 4,11,30,17,14)( 5,12,28,18,15)( 6,10,29,16,13)( 7,20,23,31,26) ( 8,21,24,32,27)( 9,19,22,33,25)$ $15, 5, 5, 5, 3$ $1782$ $15$ $( 1, 3, 2)( 4,11,29,17,14, 5,12,30,18,15, 6,10,28,16,13)( 7,20,22,33,25) ( 8,21,23,31,26)( 9,19,24,32,27)$ $10, 10, 5, 5, 2, 1$ $2673$ $10$ $( 2, 3)( 4,12,29,16,13, 5,11,30,18,14)( 6,10,28,17,15)( 7,20,24,31,26) ( 8,19,22,33,27, 9,21,23,32,25)$

## Group invariants

 Order: $26730=2 \cdot 3^{5} \cdot 5 \cdot 11$ Cyclic: no Abelian: no Solvable: yes GAP id: not available
 Character table: not available.