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Properties

Label 18T9
Order \(36\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3^2$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 2

Degree 6: $D_{6}$ x 2, $S_3^2$

Degree 9: $S_3^2$

Low degree siblings

6T9, 9T8, 12T16, 18T11 x 2

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 10]
Character table:    
     2  2  2  2  2  1  1  1  .  1
     3  2  .  1  1  1  2  1  2  2

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

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