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Properties

Label 18T63
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times S_3\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $63$
Group :  $C_2\times S_3\wr C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,6,4)(2,9,5,3)(7,14,18,12)(8,13,17,11), (1,7,12,18,10,15)(2,8,11,17,9,16)(3,5,13)(4,6,14), (1,9,12,2,10,11)(3,7,13,18,5,15)(4,8,14,17,6,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$
72:  $C_3^2:D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $S_3^2:C_2$

Low degree siblings

12T77 x 4, 12T78 x 4, 18T63 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, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 5,13)( 6,14)( 7,15)( 8,16)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 3, 9)( 4,10)( 7,14)( 8,13)(11,17)(12,18)$
$ 4, 4, 4, 4, 1, 1 $ $18$ $4$ $( 3, 9,17,11)( 4,10,18,12)( 5, 8,16,13)( 6, 7,15,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $9$ $2$ $( 3,17)( 4,18)( 5,16)( 6,15)( 7,14)( 8,13)( 9,11)(10,12)$
$ 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 $ $6$ $2$ $( 1, 2)( 3, 4)( 5,14)( 6,13)( 7,16)( 8,15)( 9,12)(10,11)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3,10)( 4, 9)( 5, 6)( 7,13)( 8,14)(11,18)(12,17)(15,16)$
$ 4, 4, 4, 4, 2 $ $18$ $4$ $( 1, 2)( 3,10,17,12)( 4, 9,18,11)( 5, 7,16,14)( 6, 8,15,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3,18)( 4,17)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 3, 6, 8,15,11)( 2, 4, 5, 7,16,12)( 9,18,13,10,17,14)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 3,18, 2, 4,17)( 5, 7, 9, 6, 8,10)(11,14,16,12,13,15)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 3,18, 2, 4,17)( 5,15, 9,14, 8,12)( 6,16,10,13, 7,11)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 4, 6, 7,15,12)( 2, 3, 5, 8,16,11)( 9,17,13)(10,18,14)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 4,18)( 2, 3,17)( 5, 8, 9)( 6, 7,10)(11,13,16)(12,14,15)$
$ 6, 6, 3, 3 $ $12$ $6$ $( 1, 4,18)( 2, 3,17)( 5,16, 9,13, 8,11)( 6,15,10,14, 7,12)$
$ 6, 6, 6 $ $4$ $6$ $( 1, 5,15, 2, 6,16)( 3, 7,11, 4, 8,12)( 9,14,17,10,13,18)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 6,15)( 2, 5,16)( 3, 8,11)( 4, 7,12)( 9,13,17)(10,14,18)$

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 186]
Character table:    
      2  4  3  3  3  4  4  3  3  3  4  2  2  2  2  2  2  2  2
      3  2  1  1  .  .  2  1  1  .  .  1  2  1  1  2  1  2  2

        1a 2a 2b 4a 2c 2d 2e 2f 4b 2g 6a 6b 6c 6d 3a 6e 6f 3b
     2P 1a 1a 1a 2c 1a 1a 1a 1a 2c 1a 3b 3a 3a 3b 3a 3a 3b 3b
     3P 1a 2a 2b 4a 2c 2d 2e 2f 4b 2g 2f 2d 2e 2b 1a 2a 2d 1a
     5P 1a 2a 2b 4a 2c 2d 2e 2f 4b 2g 6a 6b 6c 6d 3a 6e 6f 3b

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1  1 -1  1  1 -1 -1  1 -1  1 -1  1 -1 -1  1
X.3      1 -1 -1  1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1
X.4      1 -1  1 -1  1 -1  1 -1  1 -1 -1 -1  1  1  1 -1 -1  1
X.5      1 -1  1 -1  1  1 -1  1 -1  1  1  1 -1  1  1 -1  1  1
X.6      1  1 -1 -1  1 -1 -1  1  1 -1  1 -1 -1 -1  1  1 -1  1
X.7      1  1 -1 -1  1  1  1 -1 -1  1 -1  1  1 -1  1  1  1  1
X.8      1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1 -1  1
X.9      2  .  .  . -2 -2  .  .  .  2  . -2  .  .  2  . -2  2
X.10     2  .  .  . -2  2  .  .  . -2  .  2  .  .  2  .  2  2
X.11     4 -2  .  .  . -4  2  .  .  .  . -1 -1  .  1  1  2 -2
X.12     4 -2  .  .  .  4 -2  .  .  .  .  1  1  .  1  1 -2 -2
X.13     4  . -2  .  . -4  .  2  .  . -1  2  .  1 -2  . -1  1
X.14     4  . -2  .  .  4  . -2  .  .  1 -2  .  1 -2  .  1  1
X.15     4  .  2  .  . -4  . -2  .  .  1  2  . -1 -2  . -1  1
X.16     4  .  2  .  .  4  .  2  .  . -1 -2  . -1 -2  .  1  1
X.17     4  2  .  .  . -4 -2  .  .  .  . -1  1  .  1 -1  2 -2
X.18     4  2  .  .  .  4  2  .  .  .  .  1 -1  .  1 -1 -2 -2