Properties

Label 18T40
Order \(72\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:S_4$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $40$
Group :  $C_3:S_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,2,17)(3,15,4,16)(5,14,6,13)(9,12)(10,11), (1,2)(3,6)(4,5)(7,17)(8,18)(9,15)(10,16)(11,13)(12,14), (1,5,3)(2,6,4)(7,11,10)(8,12,9)(13,17,16)(14,18,15)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$ x 4
18:  $C_3^2:C_2$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 4

Degree 6: $S_4$

Degree 9: $C_3^2:C_2$

Low degree siblings

12T44 x 3, 18T37

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [72, 43]
Character table:   
     2  3  3  2  2  2  2  .  .  .
     3  2  1  .  .  1  2  2  2  2

       1a 2a 4a 2b 6a 3a 3b 3c 3d
    2P 1a 1a 2a 1a 3a 3a 3b 3c 3d
    3P 1a 2a 4a 2b 2a 1a 1a 1a 1a
    5P 1a 2a 4a 2b 6a 3a 3b 3c 3d

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