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Properties

Label 18T65
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times S_4$

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $65$
Group :  $S_3\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,16,4,10,17)(2,12,15,3,9,18)(5,8,14,6,7,13), (1,14,4,15,5,18,2,13,3,16,6,17)(7,11,10,8,12,9), (1,15)(2,16)(3,13)(4,14)(5,17)(6,18)(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 2
12:  $D_{6}$ x 2
24:  $S_4$
36:  $S_3^2$
48:  $S_4\times C_2$

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

12T83, 18T69, 18T70, 18T72

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 183]
Character table:    
      2  4  4  3  4  3  3  4  3  3  3  2   2  .  1  1
      3  2  1  1  .  .  1  1  .  1  2  1   1  2  1  2

        1a 2a 2b 2c 4a 4b 2d 2e 6a 3a 6b 12a 3b 6c 3c
     2P 1a 1a 1a 1a 2a 2a 1a 1a 3a 3a 3a  6a 3b 3c 3c
     3P 1a 2a 2b 2c 4a 4b 2d 2e 2a 1a 2b  4b 1a 2d 1a
     5P 1a 2a 2b 2c 4a 4b 2d 2e 6a 3a 6b 12a 3b 6c 3c
     7P 1a 2a 2b 2c 4a 4b 2d 2e 6a 3a 6b 12a 3b 6c 3c
    11P 1a 2a 2b 2c 4a 4b 2d 2e 6a 3a 6b 12a 3b 6c 3c

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