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Properties

Label 18T57
Order \(108\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3.S_3^2$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $57$
Group :  $C_3.S_3^2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11)(2,12)(3,13)(4,14)(15,17)(16,18), (1,7,14,2,8,13)(3,5,16,17,10,11)(4,6,15,18,9,12)
$|\Aut(F/K)|$:  $6$

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
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

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

Low degree siblings

9T18 x 2, 18T51 x 2, 18T55 x 2, 18T56, 18T57

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

Group invariants

Order:  $108=2^{2} \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [108, 17]
Character table:    
      2  2  1  2  2  1  2  1  1  .  1  1
      3  3  2  1  1  1  1  3  1  2  2  1

        1a 3a 2a 2b 6a 2c 3b 6b 3c 3d 6c
     2P 1a 3a 1a 1a 3a 1a 3b 3b 3c 3d 3d
     3P 1a 1a 2a 2b 2c 2c 1a 2a 1a 1a 2b
     5P 1a 3a 2a 2b 6a 2c 3b 6b 3c 3d 6c

X.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
X.3      1  1 -1  1 -1 -1  1 -1  1  1  1
X.4      1  1  1 -1 -1 -1  1  1  1  1 -1
X.5      2  2  . -2  .  .  2  . -1 -1  1
X.6      2  2  .  2  .  .  2  . -1 -1 -1
X.7      2 -1  .  . -1  2  2  . -1  2  .
X.8      2 -1  .  .  1 -2  2  . -1  2  .
X.9      4 -2  .  .  .  .  4  .  1 -2  .
X.10     6  . -2  .  .  . -3  1  .  .  .
X.11     6  .  2  .  .  . -3 -1  .  .  .