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Properties

Label 18T177
Order \(576\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $177$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,9,4,13,11,6,15,8)(2,18,10,3,14,12,5,16,7), (1,4,5)(2,3,6)(7,9,11)(8,10,12)(13,16,18)(14,15,17)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
9:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $C_9$

Low degree siblings

12T166 x 7, 18T177 x 6

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 $ $9$ $2$ $( 1, 2)( 3, 4)( 9,10)(11,12)(15,16)(17,18)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 3, 4)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 5, 6)(15,16)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,12,10)( 8,11, 9)(13,17,15)(14,18,16)$
$ 9, 9 $ $64$ $9$ $( 1,17, 9, 4,13,11, 6,15, 8)( 2,18,10, 3,14,12, 5,16, 7)$
$ 9, 9 $ $64$ $9$ $( 1,13, 8, 4,15, 9, 6,17,11)( 2,14, 7, 3,16,10, 5,18,12)$
$ 9, 9 $ $64$ $9$ $( 1,15,11, 4,17, 8, 6,13, 9)( 2,16,12, 3,18, 7, 5,14,10)$
$ 9, 9 $ $64$ $9$ $( 1, 9,13, 6, 8,17, 4,11,15)( 2,10,14, 5, 7,18, 3,12,16)$
$ 9, 9 $ $64$ $9$ $( 1,11,17, 6, 9,15, 4, 8,13)( 2,12,18, 5,10,16, 3, 7,14)$
$ 9, 9 $ $64$ $9$ $( 1, 8,15, 6,11,13, 4, 9,17)( 2, 7,16, 5,12,14, 3,10,18)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8661]
Character table:    
      2  6  6  6  6  6  6  6  6  .  .  .  .  .  .  .  .
      3  2  .  .  .  .  .  .  .  2  2  2  2  2  2  2  2

        1a 2a 2b 2c 2d 2e 2f 2g 3a 3b 9a 9b 9c 9d 9e 9f
     2P 1a 1a 1a 1a 1a 1a 1a 1a 3b 3a 9d 9f 9e 9b 9a 9c
     3P 1a 2a 2b 2c 2d 2e 2f 2g 1a 1a 3a 3a 3a 3b 3b 3b
     5P 1a 2a 2b 2c 2d 2e 2f 2g 3b 3a 9e 9d 9f 9a 9c 9b
     7P 1a 2a 2b 2c 2d 2e 2f 2g 3a 3b 9c 9a 9b 9e 9f 9d

X.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  A  A  A /A /A /A
X.3      1  1  1  1  1  1  1  1  1  1 /A /A /A  A  A  A
X.4      1  1  1  1  1  1  1  1  A /A  B  C  D /D /C /B
X.5      1  1  1  1  1  1  1  1  A /A  C  D  B /B /D /C
X.6      1  1  1  1  1  1  1  1  A /A  D  B  C /C /B /D
X.7      1  1  1  1  1  1  1  1 /A  A /B /C /D  D  C  B
X.8      1  1  1  1  1  1  1  1 /A  A /D /B /C  C  B  D
X.9      1  1  1  1  1  1  1  1 /A  A /C /D /B  B  D  C
X.10     9  5  1  1  1 -3 -3 -3  .  .  .  .  .  .  .  .
X.11     9  1  1 -3 -3 -3  5  1  .  .  .  .  .  .  .  .
X.12     9 -3  1 -3  5  1  1 -3  .  .  .  .  .  .  .  .
X.13     9 -3  5  1 -3  1 -3  1  .  .  .  .  .  .  .  .
X.14     9 -3 -3  5  1 -3  1  1  .  .  .  .  .  .  .  .
X.15     9  1 -3 -3  1  1 -3  5  .  .  .  .  .  .  .  .
X.16     9  1 -3  1 -3  5  1 -3  .  .  .  .  .  .  .  .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(9)^2-E(9)^5
C = E(9)^5
D = E(9)^2