Properties

Label 18T50
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $S_3\times D_9$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $50$
Group :  $S_3\times D_9$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,13,2,15,3,14)(4,11,5,10,6,12)(7,17,8,16,9,18), (1,4)(2,6)(3,5)(8,9)(10,11)(13,17)(14,16)(15,18)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 9: None

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $3$ $(10,11,12)(13,14,15)(16,17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $27$ $2$ $( 2, 3)( 4, 8)( 5, 7)( 6, 9)(10,16)(11,18)(12,17)(13,15)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,18,17)$
$ 9, 9 $ $4$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,13,17,11,14,18,12,15,16)$
$ 9, 9 $ $4$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,14,16,11,15,17,12,13,18)$
$ 9, 9 $ $2$ $9$ $( 1, 4, 9, 2, 5, 7, 3, 6, 8)(10,15,18,11,13,16,12,14,17)$
$ 9, 9 $ $2$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,13,17,11,14,18,12,15,16)$
$ 9, 9 $ $4$ $9$ $( 1, 5, 8, 2, 6, 9, 3, 4, 7)(10,14,16,11,15,17,12,13,18)$
$ 9, 9 $ $2$ $9$ $( 1, 6, 7, 2, 4, 8, 3, 5, 9)(10,14,16,11,15,17,12,13,18)$
$ 18 $ $6$ $18$ $( 1,10, 4,15, 9,18, 2,11, 5,13, 7,16, 3,12, 6,14, 8,17)$
$ 18 $ $6$ $18$ $( 1,10, 5,13, 8,17, 2,11, 6,14, 9,18, 3,12, 4,15, 7,16)$
$ 18 $ $6$ $18$ $( 1,10, 6,14, 7,16, 2,11, 4,15, 8,17, 3,12, 5,13, 9,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1,10)( 2,12)( 3,11)( 4,17)( 5,16)( 6,18)( 7,13)( 8,15)( 9,14)$
$ 6, 6, 6 $ $18$ $6$ $( 1,10, 2,12, 3,11)( 4,17, 5,16, 6,18)( 7,13, 8,15, 9,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,10)( 8,11)( 9,12)$
$ 6, 6, 6 $ $6$ $6$ $( 1,13, 2,14, 3,15)( 4,16, 5,17, 6,18)( 7,10, 8,11, 9,12)$

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 16]
Character table:   
      2  2  .  2  1  1  .  .  1  1  .  1   1   1   1  2  1  2  1
      3  3  3  .  3  3  3  3  3  3  3  3   2   2   2  1  1  2  2

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

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 -1 -1 -1 -1 -1 -1  -1  -1  -1  .  .  2  2
X.6      2  2  .  2  2 -1 -1 -1 -1 -1 -1   1   1   1  .  . -2 -2
X.7      2 -1  .  2 -1 -1 -1  2  2 -1  2   .   .   . -2  1  .  .
X.8      2 -1  .  2 -1 -1 -1  2  2 -1  2   .   .   .  2 -1  .  .
X.9      2 -1  . -1  2  A  C  B  C  B  A  -C  -A  -B  .  . -2  1
X.10     2 -1  . -1  2  B  A  C  A  C  B  -A  -B  -C  .  . -2  1
X.11     2 -1  . -1  2  C  B  A  B  A  C  -B  -C  -A  .  . -2  1
X.12     2 -1  . -1  2  A  C  B  C  B  A   C   A   B  .  .  2 -1
X.13     2 -1  . -1  2  B  A  C  A  C  B   A   B   C  .  .  2 -1
X.14     2 -1  . -1  2  C  B  A  B  A  C   B   C   A  .  .  2 -1
X.15     4 -2  .  4 -2  1  1 -2 -2  1 -2   .   .   .  .  .  .  .
X.16     4  1  . -2 -2 -B -A  D  F -C  E   .   .   .  .  .  .  .
X.17     4  1  . -2 -2 -A -C  E  D -B  F   .   .   .  .  .  .  .
X.18     4  1  . -2 -2 -C -B  F  E -A  D   .   .   .  .  .  .  .

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