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Properties

Label 18T13
Degree $18$
Order $36$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{18}$

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$18$:  $D_{9}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: $D_{9}$

Low degree siblings

18T13

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

Group invariants

Order:  $36=2^{2} \cdot 3^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [36, 4]
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 9a 18a 9b 18b 6a 3a 18c 9c
     2P 1a 1a 1a 1a 9b  9b 9c  9c 3a 3a  9a 9a
     3P 1a 2a 2b 2c 3a  6a 3a  6a 2b 1a  6a 3a
     5P 1a 2a 2b 2c 9c 18c 9a 18a 6a 3a 18b 9b
     7P 1a 2a 2b 2c 9b 18b 9c 18c 6a 3a 18a 9a
    11P 1a 2a 2b 2c 9b 18b 9c 18c 6a 3a 18a 9a
    13P 1a 2a 2b 2c 9c 18c 9a 18a 6a 3a 18b 9b
    17P 1a 2a 2b 2c 9a 18a 9b 18b 6a 3a 18c 9c

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

A = E(9)^2+E(9)^7
B = E(9)^4+E(9)^5
C = -E(9)^2-E(9)^4-E(9)^5-E(9)^7