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Properties

Label 18T66
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $66$
Group :  $C_2\times C_3:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(15,17)(16,18), (1,10,16,2,9,15)(3,11,17,4,12,18)(5,7,13,6,8,14), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,11)(8,12)(9,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$
24:  $S_4$
36:  18T12
48:  $S_4\times C_2$
72:  12T44

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 6: $S_4\times C_2$

Degree 9: $C_3^2:C_2$

Low degree siblings

18T66

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 189]
Character table:    
      2  4  4  4  3  3  4  3  3  3  3  3  3  1  1  1  1  1  1
      3  2  1  1  .  .  2  .  .  2  1  1  2  2  2  2  2  2  2

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

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