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Properties

Label 18T67
Order \(144\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_2^2:D_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $67$
Group :  $C_2\times C_2^2:D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,2,17)(3,16,4,15)(5,14,6,13)(7,9)(8,10), (1,14,8,4,16,9,6,17,12,2,13,7,3,15,10,5,18,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
18:  $D_{9}$
24:  $S_4$
36:  $D_{18}$
48:  $S_4\times C_2$
72:  18T38

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $D_{9}$

Low degree siblings

18T67

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

Group invariants

Order:  $144=2^{4} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [144, 109]
Character table:    
      2  4  4  4  3  3  4  3  3  3  3  3  3   1  1  1   1  1   1
      3  2  1  1  .  .  2  .  .  1  2  1  2   2  2  2   2  2   2

        1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18a 9a 9b 18b 9c 18c
     2P 1a 1a 1a 2b 1a 1a 2b 1a 3a 3a 3a 3a  9b 9b 9c  9c 9a  9a
     3P 1a 2a 2b 4a 2c 2d 4b 2e 2a 1a 2b 2d  6c 3a 3a  6c 3a  6c
     5P 1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18c 9c 9a 18a 9b 18b
     7P 1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18b 9b 9c 18c 9a 18a
    11P 1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18b 9b 9c 18c 9a 18a
    13P 1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18c 9c 9a 18a 9b 18b
    17P 1a 2a 2b 4a 2c 2d 4b 2e 6a 3a 6b 6c 18a 9a 9b 18b 9c 18c

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

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