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Properties

Label 18T269
Order \(1152\)
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 $t$ :  $269$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,6,2,5)(3,4)(7,9)(8,10)(13,16)(14,15), (1,10,15)(2,9,16)(3,8,18,5,12,14)(4,7,17,6,11,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$
24:  $S_4$
72:  12T45
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

12T205, 18T270

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $( 1, 2)( 5, 6)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $18$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $6$ $2$ $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(17,18)$
$ 3, 3, 3, 3, 3, 3 $ $128$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,11, 9)( 8,12,10)(13,17,16)(14,18,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $24$ $2$ $( 3, 5)( 4, 6)( 7,11)( 8,12)(13,17)(14,18)$
$ 4, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $72$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,11)( 8,12)(13,17)(14,18)$
$ 4, 4, 2, 2, 2, 2, 1, 1 $ $72$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,12, 8,11)( 9,10)(13,17)(14,18)$
$ 4, 4, 4, 2, 2, 2 $ $24$ $4$ $( 1, 2)( 3, 6, 4, 5)( 7,12, 8,11)( 9,10)(13,17,14,18)(15,16)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,15,10)( 2,16, 9)( 3,18,12)( 4,17,11)( 5,14, 8)( 6,13, 7)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,15,10, 2,16, 9)( 3,18,12)( 4,17,11)( 5,14, 8, 6,13, 7)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1,14,12)( 2,13,11)( 3,15, 8)( 4,16, 7)( 5,18,10)( 6,17, 9)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,14,12, 2,13,11)( 3,15, 8)( 4,16, 7)( 5,18,10, 6,17, 9)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,15,10)( 2,16, 9)( 3,14,12, 5,18, 8)( 4,13,11, 6,17, 7)$
$ 12, 6 $ $96$ $12$ $( 1,15,10, 2,16, 9)( 3,14,12, 6,17, 7, 4,13,11, 5,18, 8)$
$ 3, 3, 3, 3, 3, 3 $ $16$ $3$ $( 1,10,15)( 2, 9,16)( 3,12,18)( 4,11,17)( 5, 8,14)( 6, 7,13)$
$ 6, 6, 3, 3 $ $48$ $6$ $( 1,10,15, 2, 9,16)( 3,12,18)( 4,11,17)( 5, 8,14, 6, 7,13)$
$ 3, 3, 3, 3, 3, 3 $ $32$ $3$ $( 1, 8,18)( 2, 7,17)( 3,10,14)( 4, 9,13)( 5,12,15)( 6,11,16)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1, 8,18, 2, 7,17)( 3,10,14)( 4, 9,13)( 5,12,15, 6,11,16)$
$ 6, 6, 3, 3 $ $96$ $6$ $( 1,10,15)( 2, 9,16)( 3, 8,18, 5,12,14)( 4, 7,17, 6,11,13)$
$ 12, 6 $ $96$ $12$ $( 1,10,15, 2, 9,16)( 3, 8,18, 6,11,13, 4, 7,17, 5,12,14)$

Group invariants

Order:  $1152=2^{7} \cdot 3^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.