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Properties

Label 18T37
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 $t$ :  $37$
Group :  $C_3:S_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,3,2,5,4)(7,12,10,8,11,9)(13,18,16)(14,17,15), (1,6)(2,5)(3,4)(7,16,8,15)(9,14,10,13)(11,18,12,17), (1,16)(2,15)(3,13)(4,14)(5,18)(6,17)(7,11)(8,12)
$|\Aut(F/K)|$:  $2$

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, 18T40

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

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