Properties

Label 18T27
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_3.C_2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $27$
Group :  $C_2\times C_3:S_3.C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,18,6)(2,14,17,5)(3,4)(7,9,11,15)(8,10,12,16), (1,15,9,14)(2,16,10,13)(3,17,7,6)(4,18,8,5)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:C_4$

Low degree siblings

12T40 x 2, 12T41 x 2, 18T27

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 45]
Character table:   
      2  3  3  3  3  3  3  3  3  1  1  1  1
      3  2  .  .  .  2  .  .  .  2  2  2  2

        1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b
     2P 1a 2a 2a 1a 1a 2a 2a 1a 3a 3a 3b 3b
     3P 1a 4b 4a 2a 2b 4d 4c 2c 2b 1a 1a 2b
     5P 1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b

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

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