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Properties

Label 18T53
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3^3:C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $53$
Group :  $C_3^3:C_2^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3)(4,11,6,10,5,12)(7,13,9,15,8,14)(16,17,18), (1,18,7,11,15,5)(2,17,8,10,13,4)(3,16,9,12,14,6)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$, $S_3^2$

Degree 9: None

Low degree siblings

12T71, 18T53 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, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 7,15)( 8,13)( 9,14)(10,17)(11,18)(12,16)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $( 4,10,17)( 5,11,18)( 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 6, 6, 3, 3 $ $18$ $6$ $( 1, 2, 3)( 4, 5, 6)( 7,13, 9,15, 8,14)(10,18,12,17,11,16)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 2, 3)( 4,11,16)( 5,12,17)( 6,10,18)( 7, 8, 9)(13,14,15)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 3, 2)( 4,12,18)( 5,10,16)( 6,11,17)( 7, 9, 8)(13,15,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 6)( 3, 5)( 7,10)( 8,12)( 9,11)(13,16)(14,18)(15,17)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 4)( 2, 6)( 3, 5)( 7,17)( 8,16)( 9,18)(10,15)(11,14)(12,13)$
$ 6, 6, 6 $ $18$ $6$ $( 1, 4, 7,10,15,17)( 2, 6, 8,12,13,16)( 3, 5, 9,11,14,18)$
$ 6, 6, 6 $ $18$ $6$ $( 1, 4, 7,17,15,10)( 2, 6, 8,16,13,12)( 3, 5, 9,18,14,11)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,15)( 2, 8,13)( 3, 9,14)( 4,10,17)( 5,11,18)( 6,12,16)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 7,15)( 2, 8,13)( 3, 9,14)( 4,17,10)( 5,18,11)( 6,16,12)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 8,14)( 2, 9,15)( 3, 7,13)( 4,11,16)( 5,12,17)( 6,10,18)$
$ 3, 3, 3, 3, 3, 3 $ $4$ $3$ $( 1, 8,14)( 2, 9,15)( 3, 7,13)( 4,18,12)( 5,16,10)( 6,17,11)$

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 40]
Character table:    
      2  2  2  .  1  1  .  .  2  2  1  1  1  1  .  .
      3  3  1  3  3  1  3  3  1  1  1  1  3  3  3  3

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

X.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
X.3      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
X.5      2  . -1  2  . -1 -1  . -2  .  1  2 -1  2 -1
X.6      2  . -1  2  . -1 -1  .  2  . -1  2 -1  2 -1
X.7      2 -2  2 -1  1 -1 -1  .  .  .  .  2  2 -1 -1
X.8      2  . -1  2  . -1 -1 -2  .  1  . -1  2 -1  2
X.9      2  . -1  2  . -1 -1  2  . -1  . -1  2 -1  2
X.10     2  2  2 -1 -1 -1 -1  .  .  .  .  2  2 -1 -1
X.11     4  .  1  4  .  1  1  .  .  .  . -2 -2 -2 -2
X.12     4  . -2 -2  .  1  1  .  .  .  .  4 -2 -2  1
X.13     4  . -2 -2  .  1  1  .  .  .  . -2  4  1 -2
X.14     4  .  1 -2  .  A /A  .  .  .  . -2 -2  1  1
X.15     4  .  1 -2  . /A  A  .  .  .  . -2 -2  1  1

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