Properties

Label 18T45
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times D_9:C_3$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $45$
Group :  $C_2\times D_9:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,6,10,17,15,12)(4,5,9,18,16,11)(7,13)(8,14), (1,18,14,5,8,11)(2,17,13,6,7,12)(3,4)(9,15)(10,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
36:  $C_6\times S_3$
54:  $(C_9:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $(C_9:C_3):C_2$

Low degree siblings

18T45

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 26]
Character table:   
      2  2   2   2   2   2  2  2   2   2   2   2  2  1   1   1   1   1   1  1
      3  3   1   2   1   2  1  3   1   2   1   2  1  2   2   2   2   2   2  3

        1a  6a  3a  6b  3b 2a 2b  6c  6d  6e  6f 2c 9a  9b  9c 18a 18b 18c 6g
     2P 1a  3a  3b  3b  3a 1a 1a  3a  3b  3b  3a 1a 9a  9c  9b  9a  9c  9b 3c
     3P 1a  2a  1a  2a  1a 2a 2b  2c  2b  2c  2b 2c 3c  3c  3c  6g  6g  6g 2b
     5P 1a  6b  3b  6a  3a 2a 2b  6e  6f  6c  6d 2c 9a  9c  9b 18a 18c 18b 6g
     7P 1a  6a  3a  6b  3b 2a 2b  6c  6d  6e  6f 2c 9a  9b  9c 18a 18b 18c 6g
    11P 1a  6b  3b  6a  3a 2a 2b  6e  6f  6c  6d 2c 9a  9c  9b 18a 18c 18b 6g
    13P 1a  6a  3a  6b  3b 2a 2b  6c  6d  6e  6f 2c 9a  9b  9c 18a 18b 18c 6g
    17P 1a  6b  3b  6a  3a 2a 2b  6e  6f  6c  6d 2c 9a  9c  9b 18a 18c 18b 6g

X.1      1   1   1   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  -1  -1  -1 -1
X.3      1  -1   1  -1   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  -1  -1  -1 -1
X.5      1   A -/A  /A  -A -1 -1  -A  /A -/A   A  1  1 -/A  -A  -1  /A   A -1
X.6      1  /A  -A   A -/A -1 -1 -/A   A  -A  /A  1  1  -A -/A  -1   A  /A -1
X.7      1   A -/A  /A  -A -1  1   A -/A  /A  -A -1  1 -/A  -A   1 -/A  -A  1
X.8      1  /A  -A   A -/A -1  1  /A  -A   A -/A -1  1  -A -/A   1  -A -/A  1
X.9      1 -/A  -A  -A -/A  1 -1  /A   A   A  /A -1  1  -A -/A  -1   A  /A -1
X.10     1  -A -/A -/A  -A  1 -1   A  /A  /A   A -1  1 -/A  -A  -1  /A   A -1
X.11     1 -/A  -A  -A -/A  1  1 -/A  -A  -A -/A  1  1  -A -/A   1  -A -/A  1
X.12     1  -A -/A -/A  -A  1  1  -A -/A -/A  -A  1  1 -/A  -A   1 -/A  -A  1
X.13     2   .   2   .   2  .  2   .   2   .   2  . -1  -1  -1  -1  -1  -1  2
X.14     2   .   2   .   2  . -2   .  -2   .  -2  . -1  -1  -1   1   1   1 -2
X.15     2   .   B   .  /B  .  2   .   B   .  /B  . -1   A  /A  -1   A  /A  2
X.16     2   .  /B   .   B  .  2   .  /B   .   B  . -1  /A   A  -1  /A   A  2
X.17     2   .   B   .  /B  . -2   .  -B   . -/B  . -1   A  /A   1  -A -/A -2
X.18     2   .  /B   .   B  . -2   . -/B   .  -B  . -1  /A   A   1 -/A  -A -2
X.19     6   .   .   .   .  .  6   .   .   .   .  .  .   .   .   .   .   . -3
X.20     6   .   .   .   .  . -6   .   .   .   .  .  .   .   .   .   .   .  3

      2  1
      3  3

        3c
     2P 3c
     3P 1a
     5P 3c
     7P 3c
    11P 3c
    13P 3c
    17P 3c

X.1      1
X.2      1
X.3      1
X.4      1
X.5      1
X.6      1
X.7      1
X.8      1
X.9      1
X.10     1
X.11     1
X.12     1
X.13     2
X.14     2
X.15     2
X.16     2
X.17     2
X.18     2
X.19    -3
X.20    -3

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 2*E(3)
  = -1+Sqrt(-3) = 2b3