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Properties

Label 18T15
Order \(54\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times He_3$

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $15$
Group :  $C_2\times He_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,7,12,2,8,11)(3,9,14,4,10,13)(5,16,18,6,15,17), (1,2)(3,4)(5,9,7,6,10,8)(11,13,15,12,14,16)(17,18)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$ x 4
6:  $C_6$ x 4
9:  $C_3^2$
18:  $C_6 \times C_3$
27:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 9: $C_3^2:C_3$

Low degree siblings

18T15 x 3

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

Group invariants

Order:  $54=2 \cdot 3^{3}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [54, 10]
Character table: Data not available.