Properties

Label 18T12
Order \(36\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_3:S_3$

Related objects

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Group action invariants

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

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:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$ x 4

Degree 6: $D_{6}$ x 4

Degree 9: $C_3^2:C_2$

Low degree siblings

18T12

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [36, 13]
Character table:   
      2  2  2  2  2  1  1  1  1  1  1  1  1
      3  2  .  2  .  2  2  2  2  2  2  2  2

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

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      2  .  2  .  2  2 -1 -1 -1 -1 -1 -1
X.6      2  . -2  . -2  2  1 -1 -1  1  1 -1
X.7      2  .  2  . -1 -1  2  2 -1 -1 -1 -1
X.8      2  . -2  .  1 -1 -2  2 -1  1  1 -1
X.9      2  . -2  .  1 -1  1 -1 -1  1 -2  2
X.10     2  . -2  .  1 -1  1 -1  2 -2  1 -1
X.11     2  .  2  . -1 -1 -1 -1 -1 -1  2  2
X.12     2  .  2  . -1 -1 -1 -1  2  2 -1 -1