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Properties

Label 18T175
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 $t$ :  $175$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,18,10)(2,17,9)(3,6,11,7,13,15)(4,5,12,8,14,16), (1,6,18,15,9,7)(2,5,17,16,10,8)(3,12,14,4,11,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $S_3$, $C_6$ x 3
12:  $D_{6}$, $C_6\times C_2$
18:  $S_3\times C_3$
36:  $C_6\times S_3$
288:  $A_4\wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: None

Degree 9: $S_3\times C_3$

Low degree siblings

12T158, 16T1028, 18T175

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 $ $6$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 3, 4)( 9,10)(13,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,10,18)( 2, 9,17)( 3,11,13)( 4,12,14)( 5, 8,16)( 6, 7,15)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,18,10)( 2,17, 9)( 3,13,11)( 4,14,12)( 5,16, 8)( 6,15, 7)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,12,16)( 2,11,15)( 3, 7,17)( 4, 8,18)( 5, 9,14)( 6,10,13)$
$ 6, 6, 3, 3 $ $24$ $6$ $( 1,14, 8)( 2,13, 7)( 3,15, 9, 4,16,10)( 5,17,11, 6,18,12)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1,14, 8)( 2,13, 7)( 3,16, 9)( 4,15,10)( 5,18,11)( 6,17,12)$
$ 6, 6, 3, 3 $ $24$ $6$ $( 1, 4, 5, 2, 3, 6)( 7, 9,12, 8,10,11)(13,15,17)(14,16,18)$
$ 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,15,17)(14,16,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $12$ $2$ $( 3, 7)( 4, 8)( 5,14)( 6,13)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $36$ $4$ $( 1, 2)( 3, 8)( 4, 7)( 5,14, 6,13)( 9,10)(11,15,12,16)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,10,18)( 2, 9,17)( 3,15,13, 7,11, 6)( 4,16,14, 8,12, 5)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,18,10)( 2,17, 9)( 3, 6,11, 7,13,15)( 4, 5,12, 8,14,16)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(15,16)$
$ 6, 6, 6 $ $16$ $6$ $( 1,10,18, 2, 9,17)( 3,11,13, 4,12,14)( 5, 8,16, 6, 7,15)$
$ 6, 6, 6 $ $16$ $6$ $( 1,18, 9, 2,17,10)( 3,13,12, 4,14,11)( 5,16, 7, 6,15, 8)$
$ 6, 6, 6 $ $32$ $6$ $( 1,12,16, 2,11,15)( 3, 7,17, 4, 8,18)( 5, 9,14, 6,10,13)$
$ 6, 3, 3, 3, 3 $ $24$ $6$ $( 1,14, 7, 2,13, 8)( 3,15,10)( 4,16, 9)( 5,17,12)( 6,18,11)$
$ 6, 6, 6 $ $8$ $6$ $( 1,14, 7, 2,13, 8)( 3,16,10, 4,15, 9)( 5,18,12, 6,17,11)$
$ 6, 6, 6 $ $8$ $6$ $( 1, 4, 5, 2, 3, 6)( 7, 9,12, 8,10,11)(13,16,17,14,15,18)$
$ 6, 3, 3, 3, 3 $ $24$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12)( 8, 9,11)(13,16,17,14,15,18)$
$ 4, 4, 2, 2, 2, 1, 1, 1, 1 $ $36$ $4$ $( 3, 7)( 4, 8)( 5,14, 6,13)(11,15,12,16)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $12$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5,14)( 6,13)( 9,10)(11,15)(12,16)(17,18)$
$ 6, 6, 6 $ $48$ $6$ $( 1,10,18, 2, 9,17)( 3,15,14, 7,11, 6)( 4,16,13, 8,12, 5)$
$ 6, 6, 6 $ $48$ $6$ $( 1,18, 9, 2,17,10)( 3, 6,11, 7,13,16)( 4, 5,12, 8,14,15)$

Group invariants

Order:  $576=2^{6} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [576, 8656]
Character table: Data not available.