Properties

Label 18T10
Order \(36\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^2 : C_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $10$
Group :  $C_3^2 : C_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10,15)(2,14,9,16)(3,5,7,18)(4,6,8,17)(11,12), (1,14)(2,13)(3,12)(4,11)(5,9)(6,10)(15,18)(16,17)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$ x 2

Degree 9: $C_3^2:C_4$

Low degree siblings

6T10 x 2, 9T9, 12T17 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 $ $9$ $2$ $( 3,17)( 4,18)( 5,15)( 6,16)( 7,14)( 8,13)( 9,11)(10,12)$
$ 4, 4, 4, 4, 2 $ $9$ $4$ $( 1, 2)( 3, 9,17,11)( 4,10,18,12)( 5, 7,15,14)( 6, 8,16,13)$
$ 4, 4, 4, 4, 2 $ $9$ $4$ $( 1, 2)( 3,11,17, 9)( 4,12,18,10)( 5,14,15, 7)( 6,13,16, 8)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3,17)( 2, 4,18)( 5, 8, 9)( 6, 7,10)(11,13,15)(12,14,16)$
$ 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:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 9]
Character table:   
     2  2  2  2  2  .  .
     3  2  .  .  .  2  2

       1a 2a 4a 4b 3a 3b
    2P 1a 1a 2a 2a 3a 3b
    3P 1a 2a 4b 4a 1a 1a

X.1     1  1  1  1  1  1
X.2     1  1 -1 -1  1  1
X.3     1 -1  A -A  1  1
X.4     1 -1 -A  A  1  1
X.5     4  .  .  .  1 -2
X.6     4  .  .  . -2  1

A = -E(4)
  = -Sqrt(-1) = -i