Properties

 Label 18T2 Order $$18$$ n $$18$$ Cyclic No Abelian Yes Solvable Yes Primitive No $p$-group No Group: $C_6 \times C_3$

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Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $2$ Group : $C_6 \times C_3$ Parity: $-1$ Primitive: No Nilpotency class: $1$ Generators: (1,7,13,2,8,14)(3,9,15,4,10,16)(5,11,17,6,12,18), (1,16,5,2,15,6)(3,11,8,4,12,7)(9,17,14,10,18,13) $|\Aut(F/K)|$: $18$

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:  $C_3^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$ x 4

Degree 6: $C_6$ x 4

Degree 9: $C_3^2$

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

 Order: $18=2 \cdot 3^{2}$ Cyclic: No Abelian: Yes Solvable: Yes GAP id: [18, 5]
 Character table:  2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1a 2a 3a 6a 3b 6b 6c 3c 6d 3d 6e 3e 3f 6f 3g 6g 3h 6h 2P 1a 1a 3h 3h 3g 3g 3f 3f 3e 3e 3d 3d 3c 3c 3b 3b 3a 3a 3P 1a 2a 1a 2a 1a 2a 2a 1a 2a 1a 2a 1a 1a 2a 1a 2a 1a 2a 5P 1a 2a 3h 6h 3g 6g 6f 3f 6e 3e 6d 3d 3c 6c 3b 6b 3a 6a X.1 1 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 1 -1 X.3 1 -1 1 -1 A -A -A A -A A -/A /A /A -/A /A -/A 1 -1 X.4 1 -1 1 -1 /A -/A -/A /A -/A /A -A A A -A A -A 1 -1 X.5 1 -1 A -A 1 -1 -A A -/A /A -A A /A -/A 1 -1 /A -/A X.6 1 -1 /A -/A 1 -1 -/A /A -A A -/A /A A -A 1 -1 A -A X.7 1 -1 A -A A -A -/A /A -1 1 -1 1 A -A /A -/A /A -/A X.8 1 -1 /A -/A /A -/A -A A -1 1 -1 1 /A -/A A -A A -A X.9 1 -1 A -A /A -/A -1 1 -A A -/A /A 1 -1 A -A /A -/A X.10 1 -1 /A -/A A -A -1 1 -/A /A -A A 1 -1 /A -/A A -A X.11 1 1 1 1 A A A A A A /A /A /A /A /A /A 1 1 X.12 1 1 1 1 /A /A /A /A /A /A A A A A A A 1 1 X.13 1 1 A A 1 1 A A /A /A A A /A /A 1 1 /A /A X.14 1 1 /A /A 1 1 /A /A A A /A /A A A 1 1 A A X.15 1 1 A A A A /A /A 1 1 1 1 A A /A /A /A /A X.16 1 1 /A /A /A /A A A 1 1 1 1 /A /A A A A A X.17 1 1 A A /A /A 1 1 A A /A /A 1 1 A A /A /A X.18 1 1 /A /A A A 1 1 /A /A A A 1 1 /A /A A A A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3