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Properties

Label 18T153
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 $t$ :  $153$
Group :  $C_3:S_3:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,4,15,5,18,2,13,3,16,6,17)(7,12,9,8,11,10), (1,14,11,2,13,12)(3,18,7,6,15,10)(4,17,8,5,16,9)
$|\Aut(F/K)|$:  $2$

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, 18T154, 18T155

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 $ $6$ $6$ $( 7,10,11, 8, 9,12)(13,18,15,14,17,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 7,13)( 8,14)( 9,15)(10,16)(11,17)(12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $27$ $2$ $( 3, 5)( 4, 6)( 9,11)(10,12)(13,16)(14,15)(17,18)$
$ 12, 2, 2, 1, 1 $ $36$ $12$ $( 3, 5)( 4, 6)( 7,13,10,18,11,15, 8,14, 9,17,12,16)$
$ 4, 4, 4, 2, 2, 1, 1 $ $18$ $4$ $( 3, 5)( 4, 6)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
$ 6, 3, 3, 2, 2, 2 $ $12$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9,11)( 8,10,12)(13,18,15,14,17,16)$
$ 4, 4, 4, 2, 2, 2 $ $18$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7,13, 8,14)( 9,15,10,16)(11,17,12,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $9$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,12)(10,11)(13,16)(14,15)(17,18)$
$ 6, 6, 2, 2, 2 $ $36$ $6$ $( 1, 2)( 3, 6)( 4, 5)( 7,13, 9,17,11,15)( 8,14,10,18,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $18$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
$ 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 $ $6$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,11, 8, 9,12)(13,16,17,14,15,18)$
$ 6, 6, 3, 3 $ $36$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,13,11,17, 9,15)( 8,14,12,18,10,16)$
$ 12, 6 $ $36$ $12$ $( 1, 4, 5, 2, 3, 6)( 7,13,12,18, 9,15, 8,14,11,17,10,16)$
$ 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 2b 2c 12a 4a 6b 4b 2d 6c 2e 3b 6d 6e 12b 3c 3d 6f
     2P 1a 1a 3a 3a 1a 1a  6a 2a 3a 2a 1a 3a 1a 3b 3b 3b  6d 3c 3d 3d
     3P 1a 2a 1a 2a 2b 2c  4a 4a 2a 4b 2d 2e 2e 1a 2a 2b  4b 1a 1a 2d
     5P 1a 2a 3a 6a 2b 2c 12a 4a 6b 4b 2d 6c 2e 3b 6d 6e 12b 3c 3d 6f
     7P 1a 2a 3a 6a 2b 2c 12a 4a 6b 4b 2d 6c 2e 3b 6d 6e 12b 3c 3d 6f
    11P 1a 2a 3a 6a 2b 2c 12a 4a 6b 4b 2d 6c 2e 3b 6d 6e 12b 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  .   .  .  .  .