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Properties

Label 18T155
Order \(432\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_3:S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $155$
Group :  $C_3:S_3:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (7,11,9)(8,12,10)(13,15,17)(14,16,18), (1,15,9)(2,16,10)(3,17,11)(4,18,12)(5,13,7)(6,14,8), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12), (1,9)(2,10)(3,7)(4,8)(5,11)(6,12)(13,17)(14,18), (1,4)(2,3)(5,6)(7,10)(8,9)(11,12)(13,18)(14,17)(15,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$ x 2
12:  $D_{6}$ x 2
24:  $S_4$
36:  $S_3^2$
48:  $S_4\times C_2$
108:  $C_3^2 : D_{6} $
144:  12T83

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 6: $S_4\times C_2$

Degree 9: $C_3^2 : D_{6} $

Low degree siblings

18T152, 18T153, 18T154

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, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $6$ $3$ $( 7, 9,11)( 8,10,12)(13,17,15)(14,18,16)$
$ 6, 6, 1, 1, 1, 1, 1, 1 $ $6$ $6$ $( 7,10,11, 8, 9,12)(13,18,15,14,17,16)$
$ 4, 4, 4, 1, 1, 1, 1, 1, 1 $ $18$ $4$ $( 7,13, 8,14)( 9,15,10,16)(11,17,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $27$ $2$ $( 3, 5)( 4, 6)( 9,11)(10,12)(13,16)(14,15)(17,18)$
$ 6, 6, 2, 2, 1, 1 $ $36$ $6$ $( 3, 5)( 4, 6)( 7,13, 9,17,11,15)( 8,14,10,18,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $18$ $2$ $( 3, 5)( 4, 6)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
$ 6, 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,11)( 8,10,12)(13,18,15,14,17,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,12)(10,11)(13,16)(14,15)(17,18)$
$ 12, 2, 2, 2 $ $36$ $12$ $( 1, 2)( 3, 6)( 4, 5)( 7,13,10,18,11,15, 8,14, 9,17,12,16)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 6)( 4, 5)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
$ 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,11)( 8,10,12)(13,15,17)(14,16,18)$
$ 6, 6, 3, 3 $ $6$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,11, 8, 9,12)(13,16,17,14,15,18)$
$ 12, 3, 3 $ $36$ $12$ $( 1, 3, 5)( 2, 4, 6)( 7,13,12,18, 9,15, 8,14,11,17,10,16)$
$ 6, 6, 6 $ $36$ $6$ $( 1, 4, 5, 2, 3, 6)( 7,13,11,17, 9,15)( 8,14,12,18,10,16)$
$ 3, 3, 3, 3, 3, 3 $ $48$ $3$ $( 1, 7,13)( 2, 8,14)( 3, 9,15)( 4,10,16)( 5,11,17)( 6,12,18)$
$ 3, 3, 3, 3, 3, 3 $ $24$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,13)( 4,10,14)( 5,11,15)( 6,12,16)$
$ 6, 6, 6 $ $72$ $6$ $( 1, 7,13, 4,12,16)( 2, 8,14, 3,11,15)( 5, 9,17, 6,10,18)$

Group invariants

Order:  $432=2^{4} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [432, 523]
Character table:    
      2  4  4  3  3  3  4  2  3  2  3  4   2  3  3  3   2  2  .  1  1
      3  3  2  2  2  1  .  1  1  2  1  1   1  1  3  2   1  1  2  2  1

        1a 2a 3a 6a 4a 2b 6b 2c 6c 2d 2e 12a 4b 3b 6d 12b 6e 3c 3d 6f
     2P 1a 1a 3a 3a 2a 1a 3a 1a 3a 1a 1a  6a 2a 3b 3b  6d 3b 3c 3d 3d
     3P 1a 2a 1a 2a 4a 2b 2c 2c 2a 2d 2e  4b 4b 1a 2a  4a 2d 1a 1a 2e
     5P 1a 2a 3a 6a 4a 2b 6b 2c 6c 2d 2e 12a 4b 3b 6d 12b 6e 3c 3d 6f
     7P 1a 2a 3a 6a 4a 2b 6b 2c 6c 2d 2e 12a 4b 3b 6d 12b 6e 3c 3d 6f
    11P 1a 2a 3a 6a 4a 2b 6b 2c 6c 2d 2e 12a 4b 3b 6d 12b 6e 3c 3d 6f

X.1      1  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  1 -1
X.3      1  1  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  1 -1
X.5      2  2  2  2  . -2  .  .  2  . -2   .  .  2  2   .  . -1 -1  1
X.6      2  2  2  2  .  2  .  .  2  .  2   .  .  2  2   .  . -1 -1 -1
X.7      2  2 -1 -1  .  .  1 -2 -1  .  .   1 -2  2  2   .  . -1  2  .
X.8      2  2 -1 -1  .  . -1  2 -1  .  .  -1  2  2  2   .  . -1  2  .
X.9      3 -1  3 -1 -1 -1  1  1 -1  1  3  -1 -1  3 -1  -1  1  .  .  .
X.10     3 -1  3 -1 -1  1 -1 -1 -1  1 -3   1  1  3 -1  -1  1  .  .  .
X.11     3 -1  3 -1  1 -1 -1 -1 -1 -1  3   1  1  3 -1   1 -1  .  .  .
X.12     3 -1  3 -1  1  1  1  1 -1 -1 -3  -1 -1  3 -1   1 -1  .  .  .
X.13     4  4 -2 -2  .  .  .  . -2  .  .   .  .  4  4   .  .  1 -2  .
X.14     6  6  .  . -2  .  .  .  . -2  .   .  . -3 -3   1  1  .  .  .
X.15     6  6  .  .  2  .  .  .  .  2  .   .  . -3 -3  -1 -1  .  .  .
X.16     6 -2 -3  1  .  . -1  2  1  .  .   1 -2  6 -2   .  .  .  .  .
X.17     6 -2 -3  1  .  .  1 -2  1  .  .  -1  2  6 -2   .  .  .  .  .
X.18     6 -2  .  4 -2  .  .  . -2  2  .   .  . -3  1   1 -1  .  .  .
X.19     6 -2  .  4  2  .  .  . -2 -2  .   .  . -3  1  -1  1  .  .  .
X.20    12 -4  . -4  .  .  .  .  2  .  .   .  . -6  2   .  .  .  .  .