Properties

Label 18T34
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\wr C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $34$
Group :  $S_3\wr C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,10)(2,11,9)(3,16,6,17,14,7)(4,15,5,18,13,8), (1,15,3,8)(2,16,4,7)(5,10,11,14)(6,9,12,13)(17,18)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:D_4$

Degree 9: $S_3^2:C_2$

Low degree siblings

6T13 x 2, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34, 18T36

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 40]
Character table:   
     2  3  2  3  2  2  1  1  1  1
     3  2  1  .  1  .  2  1  1  2

       1a 2a 2b 2c 4a 3a 6a 6b 3b
    2P 1a 1a 1a 1a 2b 3a 3a 3b 3b
    3P 1a 2a 2b 2c 4a 1a 2a 2c 1a
    5P 1a 2a 2b 2c 4a 3a 6a 6b 3b

X.1     1  1  1  1  1  1  1  1  1
X.2     1 -1  1 -1  1  1 -1 -1  1
X.3     1 -1  1  1 -1  1 -1  1  1
X.4     1  1  1 -1 -1  1  1 -1  1
X.5     2  . -2  .  .  2  .  .  2
X.6     4 -2  .  .  .  1  1  . -2
X.7     4  .  . -2  . -2  .  1  1
X.8     4  .  .  2  . -2  . -1  1
X.9     4  2  .  .  .  1 -1  . -2