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Properties

Label 18T185
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 $t$ :  $185$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,14,11,9,3)(2,17,13,12,10,4)(5,8,16,6,7,15), (1,6)(2,5)(7,13,8,14)(9,15,10,16)(11,18,12,17)
$|\Aut(F/K)|$:  $2$

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
36:  $S_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$ x 2

Degree 6: None

Degree 9: $S_3^2$

Low degree siblings

8T45, 12T161, 12T163, 12T165 x 2, 16T1032, 16T1034, 18T179, 18T180, 18T185

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 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,14, 9)( 2,13,10)( 3,18,11)( 4,17,12)( 5,16, 7)( 6,15, 8)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,16,11)( 2,15,12)( 3,14, 8)( 4,13, 7)( 5,17, 9)( 6,18,10)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,16)( 4,10,15)( 5,12,13)( 6,11,14)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1, 8,18, 2, 7,17)( 3, 9,15)( 4,10,16)( 5,11,14, 6,12,13)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ $36$ $4$ $( 3, 7)( 4, 8)( 5,17, 6,18)(11,15,12,16)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5,17)( 6,18)( 9,10)(11,15)(12,16)(13,14)$
$ 6, 6, 6 $ $96$ $6$ $( 1,14,10, 2,13, 9)( 3, 5,12, 7,18,16)( 4, 6,11, 8,17,15)$
$ 4, 4, 4, 2, 2, 1, 1 $ $72$ $4$ $( 3, 5)( 4, 6)( 7,18, 8,17)( 9,14,10,13)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $36$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7,17)( 8,18)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2, 2 $ $36$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,17, 8,18)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 3,17)( 4,18)( 5, 7)( 6, 8)( 9,13)(10,14)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 1, 2)( 3,18)( 4,17)( 5, 7, 6, 8)( 9,14,10,13)(15,16)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,16,11)( 2,15,12)( 3, 9, 7,17,14, 6)( 4,10, 8,18,13, 5)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8654]
Character table:    
      2  6  5  6  1  1  2  2  4  4  1  3  4  4  4  4  1
      3  2  1  .  2  2  2  1  .  1  1  .  .  .  1  .  1

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

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 -1 -1 -1  1  1  1
X.3      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
X.5      2  2  2 -1  2 -1 -1  2  2 -1  .  .  .  .  .  .
X.6      2  2  2 -1  2 -1 -1 -2 -2  1  .  .  .  .  .  .
X.7      2  2  2  2 -1 -1 -1  .  .  .  .  .  . -2 -2  1
X.8      2  2  2  2 -1 -1 -1  .  .  .  .  .  .  2  2 -1
X.9      4  4  4 -2 -2  1  1  .  .  .  .  .  .  .  .  .
X.10     6  2 -2  .  .  3 -1  .  .  .  . -2  2  .  .  .
X.11     6  2 -2  .  .  3 -1  .  .  .  .  2 -2  .  .  .
X.12     9 -3  1  .  .  .  . -1  3  . -1  1  1  3 -1  .
X.13     9 -3  1  .  .  .  . -1  3  .  1 -1 -1 -3  1  .
X.14     9 -3  1  .  .  .  .  1 -3  . -1  1  1 -3  1  .
X.15     9 -3  1  .  .  .  .  1 -3  .  1 -1 -1  3 -1  .
X.16    12  4 -4  .  . -3  1  .  .  .  .  .  .  .  .  .