Properties

Label 18T7
Order \(36\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2 : C_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $7$
Group :  $C_2^2 : C_9$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,16,6,11,14,3,10,17)(2,8,15,5,12,13,4,9,18), (1,17,9,3,14,12,6,16,8)(2,18,10,4,13,11,5,15,7)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$
12:  $A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 9: $C_9$

Low degree siblings

There are no siblings 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 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,11, 8,10,12)(13,16,18,14,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,10,11)( 8, 9,12)(13,15,18)(14,16,17)$
$ 6, 6, 3, 3 $ $3$ $6$ $( 1, 5, 3, 2, 6, 4)( 7,11,10)( 8,12, 9)(13,17,15,14,18,16)$
$ 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6, 3)( 2, 5, 4)( 7,11,10)( 8,12, 9)(13,18,15)(14,17,16)$
$ 9, 9 $ $4$ $9$ $( 1, 7,15, 6,11,13, 3,10,18)( 2, 8,16, 5,12,14, 4, 9,17)$
$ 9, 9 $ $4$ $9$ $( 1, 9,13, 6, 8,18, 3,12,15)( 2,10,14, 5, 7,17, 4,11,16)$
$ 9, 9 $ $4$ $9$ $( 1,11,18, 6,10,15, 3, 7,13)( 2,12,17, 5, 9,16, 4, 8,14)$
$ 9, 9 $ $4$ $9$ $( 1,13, 7, 3,15,10, 6,18,11)( 2,14, 8, 4,16, 9, 5,17,12)$
$ 9, 9 $ $4$ $9$ $( 1,15,12, 3,18, 8, 6,13, 9)( 2,16,11, 4,17, 7, 5,14,10)$
$ 9, 9 $ $4$ $9$ $( 1,17,10, 3,14,11, 6,16, 7)( 2,18, 9, 4,13,12, 5,15, 8)$

Group invariants

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

        1a 2a  6a 3a  6b 3b 9a 9b 9c 9d 9e 9f
     2P 1a 1a  3b 3b  3a 3a 9e 9d 9f 9a 9c 9b
     3P 1a 2a  2a 1a  2a 1a 3b 3b 3b 3a 3a 3a
     5P 1a 2a  6b 3b  6a 3a 9d 9f 9e 9b 9a 9c
     7P 1a 2a  6a 3a  6b 3b 9b 9c 9a 9f 9d 9e

X.1      1  1   1  1   1  1  1  1  1  1  1  1
X.2      1  1   1  1   1  1  A  A  A /A /A /A
X.3      1  1   1  1   1  1 /A /A /A  A  A  A
X.4      1  1   A  A  /A /A  C  D  E /E /D /C
X.5      1  1   A  A  /A /A  D  E  C /C /E /D
X.6      1  1   A  A  /A /A  E  C  D /D /C /E
X.7      1  1  /A /A   A  A /C /D /E  E  D  C
X.8      1  1  /A /A   A  A /E /C /D  D  C  E
X.9      1  1  /A /A   A  A /D /E /C  C  E  D
X.10     3 -1  -1  3  -1  3  .  .  .  .  .  .
X.11     3 -1 -/A  B  -A /B  .  .  .  .  .  .
X.12     3 -1  -A /B -/A  B  .  .  .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)
  = (-3+3*Sqrt(-3))/2 = 3b3
C = -E(9)^4-E(9)^7
D = E(9)^7
E = E(9)^4