# Properties

 Label 18T25 Degree $18$ Order $72$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_6\times A_4$

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## Group action invariants

 Degree $n$: $18$ Transitive number $t$: $25$ Group: $C_6\times A_4$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (1,9,15,2,10,16)(3,11,17,4,12,18)(5,7,14,6,8,13), (1,3,6,2,4,5)(7,10,11)(8,9,12)(13,16,18,14,15,17)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$18$:  $C_6 \times C_3$
$24$:  $A_4\times C_2$
$36$:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4\times C_2$

Degree 9: $C_3^2$

## 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $(13,14)(15,16)(17,18)$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ $2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ $6, 6, 3, 3$ $3$ $6$ $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,15,18)(14,16,17)$ $6, 6, 6$ $1$ $6$ $( 1, 3, 6, 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)$ $6, 3, 3, 3, 3$ $3$ $6$ $( 1, 3, 6, 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$ $3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$ $6, 6, 3, 3$ $3$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,17,15,14,18,16)$ $6, 3, 3, 3, 3$ $3$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ $6, 6, 6$ $1$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,12,10, 8,11, 9)(13,17,15,14,18,16)$ $3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,12,15)( 6,11,16)$ $6, 6, 6$ $4$ $6$ $( 1, 7,17, 2, 8,18)( 3, 9,13, 4,10,14)( 5,12,15, 6,11,16)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 9,16)( 2,10,15)( 3,11,18)( 4,12,17)( 5, 7,13)( 6, 8,14)$ $6, 6, 6$ $4$ $6$ $( 1, 9,16, 2,10,15)( 3,11,18, 4,12,17)( 5, 7,13, 6, 8,14)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,11,13)( 2,12,14)( 3, 8,16)( 4, 7,15)( 5, 9,17)( 6,10,18)$ $6, 6, 6$ $4$ $6$ $( 1,11,13, 2,12,14)( 3, 8,16, 4, 7,15)( 5, 9,17, 6,10,18)$ $6, 6, 6$ $4$ $6$ $( 1,13,11, 2,14,12)( 3,16, 8, 4,15, 7)( 5,17, 9, 6,18,10)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,17,10)( 6,18, 9)$ $6, 6, 6$ $4$ $6$ $( 1,15,10, 2,16, 9)( 3,17,12, 4,18,11)( 5,14, 8, 6,13, 7)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,15, 9)( 2,16,10)( 3,17,11)( 4,18,12)( 5,14, 7)( 6,13, 8)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,17, 7)( 2,18, 8)( 3,13, 9)( 4,14,10)( 5,15,12)( 6,16,11)$ $6, 6, 6$ $4$ $6$ $( 1,17, 8, 2,18, 7)( 3,13,10, 4,14, 9)( 5,15,11, 6,16,12)$

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [72, 47]
 Character table: not available.