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Properties

Label 18T60
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times A_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $60$
Group :  $C_2\times S_3\times A_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,15)(2,10,16)(3,8,17,6,11,14)(4,7,18,5,12,13), (1,7,17,2,8,18)(3,10,13,4,9,14)(5,12,16,6,11,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $A_4$, $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $A_4\times C_2$ x 3
36:  $C_6\times S_3$
48:  $C_2^2 \times A_4$
72:  12T43

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $A_4\times C_2$

Degree 9: $S_3\times C_3$

Low degree siblings

18T60

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 190]
Character table: Data not available.