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Properties

Label 18T52
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3^2:S_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $52$
Group :  $C_2\times C_3^2:S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,10)(2,14,9)(3,15,6)(4,16,5)(7,18,12)(8,17,11), (1,18,4,2,17,3)(5,14,8,15,10,12)(6,13,7,16,9,11), (1,16,4,11,17,13)(2,15,3,12,18,14)(5,10,8)(6,9,7)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 4
12:  $D_{6}$ x 4
18:  $C_3^2:C_2$
36:  18T12
54:  $(C_3^2:C_3):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $(C_3^2:C_3):C_2$

Low degree siblings

18T52 x 7

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 28]
Character table:    
      2  2  1  2  2  1  2   2   2   2   2   2  1  1  1   2  1  1  1   2   2
      3  3  2  1  3  2  1   3   1   3   1   1  2  2  2   1  2  2  2   3   3

        1a 3a 2a 2b 6a 2c  6b  6c  3b  6d  6e 3c 3d 3e  6f 6g 6h 6i  3f  6j
     2P 1a 3a 1a 1a 3a 1a  3f  3f  3f  3f  3b 3c 3d 3e  3b 3c 3d 3e  3b  3b
     3P 1a 1a 2a 2b 2b 2c  2b  2c  1a  2a  2a 1a 1a 1a  2c 2b 2b 2b  1a  2b
     5P 1a 3a 2a 2b 6a 2c  6j  6f  3f  6e  6d 3c 3d 3e  6c 6g 6h 6i  3b  6b

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

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