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Properties

Label 18T463
Order \(4608\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $463$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,14,4,7,16,2,9,13,3,8,15)(5,12,17,6,11,18), (1,6,14,18,8,11,2,5,13,17,7,12)(3,10,15,4,9,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $S_4$, $A_4\times C_2$
72:  12T43, 12T45
288:  $A_4\wr C_2$, 16T709
1152:  12T205, 12T206

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: None

Degree 9: $S_3\times C_3$

Low degree siblings

18T462 x 3, 18T463 x 2

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, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 5, 6)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(17,18)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 9,10)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)(11,12)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 7,13)( 2, 8,14)( 3,10,16)( 4, 9,15)( 5,11,17)( 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1,13, 7)( 2,14, 8)( 3,16,10)( 4,15, 9)( 5,17,11)( 6,18,12)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 9,10)(11,12)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)( 7, 8)( 9,10)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 7, 8)( 9,10)(11,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 3, 4)( 7, 8)(13,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 9,10)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)(11,12)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)( 5, 6)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 5, 6)(13,14)(15,16)(17,18)$
$ 6, 6, 3, 3 $ $192$ $6$ $( 1, 7,13, 2, 8,14)( 3,10,16, 4, 9,15)( 5,11,17)( 6,12,18)$
$ 6, 6, 3, 3 $ $192$ $6$ $( 1,13, 8, 2,14, 7)( 3,16, 9, 4,15,10)( 5,17,11)( 6,18,12)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,13,15)(12,14,16)$
$ 6, 6, 3, 3 $ $384$ $6$ $( 1, 4,18)( 2, 3,17)( 5, 8,10, 6, 7, 9)(11,14,15,12,13,16)$
$ 6, 6, 3, 3 $ $384$ $6$ $( 1, 9,11)( 2,10,12)( 3, 6,13, 4, 5,14)( 7,16,18, 8,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1,10,12)( 2, 9,11)( 3, 5,13)( 4, 6,14)( 7,16,17)( 8,15,18)$
$ 6, 6, 3, 3 $ $384$ $6$ $( 1,15, 5, 2,16, 6)( 3,12, 8, 4,11, 7)( 9,17,13)(10,18,14)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1,15, 5)( 2,16, 6)( 3,11, 7)( 4,12, 8)( 9,18,13)(10,17,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 3,17)( 4,18)( 5,10, 6, 9)( 7, 8)(11,16,12,15)(13,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1, 2)( 3,18, 4,17)( 5, 9)( 6,10)( 7, 8)(11,15,12,16)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 3,17, 4,18)( 5,10)( 6, 9)( 7, 8)(11,15,12,16)(13,14)$
$ 4, 4, 2, 2, 1, 1, 1, 1, 1, 1 $ $72$ $4$ $( 3,18, 4,17)( 5, 9, 6,10)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $72$ $2$ $( 1, 2)( 3,17)( 4,18)( 5,10)( 6, 9)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $24$ $2$ $( 3,18)( 4,17)( 5, 9)( 6,10)(11,16)(12,15)$
$ 6, 6, 3, 3 $ $384$ $6$ $( 1, 7,14)( 2, 8,13)( 3, 5,16,18, 9,11)( 4, 6,15,17,10,12)$
$ 6, 6, 3, 3 $ $384$ $6$ $( 1,13, 8)( 2,14, 7)( 3,11,10,18,15, 5)( 4,12, 9,17,16, 6)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 3,17)( 4,18)( 5,10, 6, 9)( 7, 8)(11,15)(12,16)$
$ 4, 2, 2, 2, 2, 2, 2, 2 $ $72$ $4$ $( 1, 2)( 3,18, 4,17)( 5, 9)( 6,10)( 7, 8)(11,16)(12,15)(13,14)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 3,17, 4,18)( 5,10)( 6, 9)( 7, 8)(11,16)(12,15)$
$ 4, 4, 4, 2, 1, 1, 1, 1 $ $72$ $4$ $( 3,18, 4,17)( 5, 9, 6,10)(11,16,12,15)(13,14)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 3,17)( 4,18)( 5, 9)( 6,10)( 7, 8)(11,16,12,15)$
$ 4, 4, 4, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3,18, 4,17)( 5,10, 6, 9)( 7, 8)(11,15,12,16)(13,14)$
$ 12, 6 $ $384$ $12$ $( 1, 7,14, 2, 8,13)( 3, 5,16,18, 9,11, 4, 6,15,17,10,12)$
$ 12, 6 $ $384$ $12$ $( 1,13, 7, 2,14, 8)( 3,11, 9,17,16, 6, 4,12,10,18,15, 5)$

Group invariants

Order:  $4608=2^{9} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.