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Properties

Label 18T69
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $69$
Group :  $S_3\times S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,2,10)(3,8,4,7)(5,11,6,12)(13,18)(14,17)(15,16), (1,6,3,2,5,4)(7,17,10,13,12,15)(8,18,9,14,11,16)
$|\Aut(F/K)|$:  $2$

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, 18T65, 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 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $6$ $4$ $( 7,13, 8,14)( 9,16,10,15)(11,18,12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $9$ $2$ $( 3, 5)( 4, 6)( 7,11)( 8,12)( 9,10)(13,17)(14,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $18$ $2$ $( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,13)( 8,14)( 9,16)(10,15)(11,18)(12,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7,11)( 8,12)( 9,10)(13,18)(14,17)(15,16)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 6)( 4, 5)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$
$ 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 $ $2$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$
$ 12, 3, 3 $ $12$ $12$ $( 1, 3, 5)( 2, 4, 6)( 7,15,11,14,10,17, 8,16,12,13, 9,18)$
$ 6, 6, 6 $ $12$ $6$ $( 1, 4, 5, 2, 3, 6)( 7,15,12,13,10,17)( 8,16,11,14, 9,18)$
$ 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 4a 2b 2c 2d 2e 4b 6a 3a 12a 6b 3b 6c 3c
     2P 1a 1a 2a 1a 1a 1a 1a 2a 3a 3a  6a 3a 3b 3c 3c
     3P 1a 2a 4a 2b 2c 2d 2e 4b 2a 1a  4a 2d 1a 2e 1a
     5P 1a 2a 4a 2b 2c 2d 2e 4b 6a 3a 12a 6b 3b 6c 3c
     7P 1a 2a 4a 2b 2c 2d 2e 4b 6a 3a 12a 6b 3b 6c 3c
    11P 1a 2a 4a 2b 2c 2d 2e 4b 6a 3a 12a 6b 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  .  .  .