# Properties

 Label 18T3 Order $$18$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3 \times C_3$

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## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $3$ Group : $S_3 \times C_3$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,9,11)(2,10,12)(3,6,13)(4,5,14)(7,16,18)(8,15,17), (1,2)(3,17)(4,18)(5,10)(6,9)(7,8)(11,16)(12,15)(13,14) $|\Aut(F/K)|$: $18$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $S_3$, $S_3\times C_3$

Degree 9: $S_3\times C_3$

## Low degree siblings

6T5, 9T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

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

## Group invariants

 Order: $18=2 \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [18, 3]
 Character table:  2 1 1 . . 1 1 . 1 1 3 2 1 2 2 1 2 2 1 2 1a 2a 3a 3b 6a 3c 3d 6b 3e 2P 1a 1a 3a 3d 3e 3e 3b 3c 3c 3P 1a 2a 1a 1a 2a 1a 1a 2a 1a 5P 1a 2a 3a 3d 6b 3e 3b 6a 3c X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 1 1 -1 1 X.3 1 -1 1 A -A A /A -/A /A X.4 1 -1 1 /A -/A /A A -A A X.5 1 1 1 A A A /A /A /A X.6 1 1 1 /A /A /A A A A X.7 2 . -1 -1 . 2 -1 . 2 X.8 2 . -1 -/A . B -A . /B X.9 2 . -1 -A . /B -/A . B A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3) = -1+Sqrt(-3) = 2b3