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Properties

Label 18T111
Order \(288\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $111$
Group :  $C_2\times S_3\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,6,15,7)(2,14,9,5,16,8)(3,18,12)(4,17,11), (1,6,3)(2,5,4)(7,18,9,14,12,15,8,17,10,13,11,16), (1,9,15,2,10,16)(3,8,18,5,12,14)(4,7,17,6,11,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$ x 2
8:  $C_2^3$
12:  $D_{6}$ x 6
24:  $S_4$, $S_3 \times C_2^2$ x 2
36:  $S_3^2$
48:  $S_4\times C_2$ x 3
72:  12T37
96:  12T48
144:  12T83

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 2

Degree 6: $S_4\times C_2$

Degree 9: $S_3^2$

Low degree siblings

18T111 x 3

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, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 7,13)( 8,14)( 9,16)(10,15)(11,17)(12,18)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $6$ $4$ $( 7,13, 8,14)( 9,16,10,15)(11,17,12,18)$
$ 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, 1, 1 $ $18$ $2$ $( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 4, 2, 2, 1, 1 $ $18$ $4$ $( 3, 6)( 4, 5)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
$ 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 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,13)( 8,14)( 9,16)(10,15)(11,17)(12,18)$
$ 4, 4, 4, 2, 2, 2 $ $6$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7,13, 8,14)( 9,16,10,15)(11,17,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7,11)( 8,12)( 9,10)(13,17)(14,18)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 5)( 4, 6)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
$ 6, 3, 3, 3, 3 $ $6$ $6$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,12, 8,10,11)(13,15,18)(14,16,17)$
$ 6, 6, 3, 3 $ $6$ $6$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,12, 8,10,11)(13,16,18,14,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,10,12)( 8, 9,11)(13,15,18)(14,16,17)$
$ 12, 3, 3 $ $12$ $12$ $( 1, 3, 6)( 2, 4, 5)( 7,15,11,14,10,18, 8,16,12,13, 9,17)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 3, 6)( 2, 4, 5)( 7,15,12,13,10,18)( 8,16,11,14, 9,17)$
$ 6, 6, 6 $ $2$ $6$ $( 1, 4, 6, 2, 3, 5)( 7, 9,12, 8,10,11)(13,16,18,14,15,17)$
$ 12, 6 $ $12$ $12$ $( 1, 4, 6, 2, 3, 5)( 7,15,11,14,10,18, 8,16,12,13, 9,17)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 4, 6, 2, 3, 5)( 7,15,12,13,10,18)( 8,16,11,14, 9,17)$
$ 6, 6, 6 $ $16$ $6$ $( 1, 7,17, 2, 8,18)( 3,10,14, 4, 9,13)( 5,11,15, 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 7,17)( 2, 8,18)( 3,10,14)( 4, 9,13)( 5,11,15)( 6,12,16)$
$ 6, 6, 3, 3 $ $24$ $6$ $( 1, 7,15, 6,10,13)( 2, 8,16, 5, 9,14)( 3,12,18)( 4,11,17)$
$ 6, 6, 6 $ $24$ $6$ $( 1, 7,15, 5, 9,14)( 2, 8,16, 6,10,13)( 3,12,18, 4,11,17)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 9,16)( 2,10,15)( 3,11,17)( 4,12,18)( 5, 7,13)( 6, 8,14)$
$ 6, 6, 6 $ $8$ $6$ $( 1, 9,16, 2,10,15)( 3,11,17, 4,12,18)( 5, 7,13, 6, 8,14)$

Group invariants

Order:  $288=2^{5} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [288, 1028]
Character table: Data not available.