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Properties

Label 18T61
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $61$
Group :  $C_6\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,6,11,4,9,2,8,5,12,3,10)(13,16,17,14,15,18), (1,15,9,2,16,10)(3,18,11,4,17,12)(5,13,7,6,14,8)
$|\Aut(F/K)|$:  $6$

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:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $S_4$
36:  $C_6\times S_3$
48:  $S_4\times C_2$
72:  12T45

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $S_3\times C_3$

Low degree siblings

18T61

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 188]
Character table: Data not available.