# Properties

 Label 18T6 Degree $18$ Order $36$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3 \times C_6$

# Related objects

## Group action invariants

 Degree $n$: $18$ Transitive number $t$: $6$ Group: $S_3 \times C_6$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (1,15,8,4,13,9)(2,16,7,3,14,10)(5,17,12)(6,18,11), (1,3,18,2,4,17)(5,8,10,6,7,9)(11,14,15,12,13,16)

## 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
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$, $S_3$

Degree 6: $C_6$, $D_{6}$

Degree 9: $S_3\times C_3$

## Low degree siblings

12T18, 18T6

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

## Group invariants

 Order: $36=2^{2} \cdot 3^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [36, 12]
 Character table:  2 2 2 2 2 1 1 1 2 1 2 2 2 1 2 1 2 2 2 3 2 1 2 1 2 2 2 1 2 1 2 2 2 1 2 1 2 2 1a 2a 2b 2c 6a 3a 6b 6c 3b 6d 6e 3c 3d 6f 6g 6h 3e 6i 2P 1a 1a 1a 1a 3a 3a 3d 3e 3d 3e 3e 3e 3b 3c 3b 3c 3c 3c 3P 1a 2a 2b 2c 2b 1a 2b 2c 1a 2a 2b 1a 1a 2a 2b 2c 1a 2b 5P 1a 2a 2b 2c 6a 3a 6g 6h 3d 6f 6i 3e 3b 6d 6b 6c 3c 6e 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 1 1 1 -1 1 -1 1 1 1 -1 1 -1 1 1 X.4 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 1 -1 X.5 1 -1 -1 1 -1 1 A -A -A A A -A -/A /A /A -/A -/A /A X.6 1 -1 -1 1 -1 1 /A -/A -/A /A /A -/A -A A A -A -A A X.7 1 -1 1 -1 1 1 -/A /A -/A /A -/A -/A -A A -A A -A -A X.8 1 -1 1 -1 1 1 -A A -A A -A -A -/A /A -/A /A -/A -/A X.9 1 1 -1 -1 -1 1 A A -A -A A -A -/A -/A /A /A -/A /A X.10 1 1 -1 -1 -1 1 /A /A -/A -/A /A -/A -A -A A A -A A X.11 1 1 1 1 1 1 -/A -/A -/A -/A -/A -/A -A -A -A -A -A -A X.12 1 1 1 1 1 1 -A -A -A -A -A -A -/A -/A -/A -/A -/A -/A X.13 2 . -2 . 1 -1 1 . -1 . -2 2 -1 . 1 . 2 -2 X.14 2 . 2 . -1 -1 -1 . -1 . 2 2 -1 . -1 . 2 2 X.15 2 . -2 . 1 -1 -/A . /A . B -B A . -A . -/B /B X.16 2 . -2 . 1 -1 -A . A . /B -/B /A . -/A . -B B X.17 2 . 2 . -1 -1 /A . /A . -B -B A . A . -/B -/B X.18 2 . 2 . -1 -1 A . A . -/B -/B /A . /A . -B -B A = -E(3) = (1-Sqrt(-3))/2 = -b3 B = -2*E(3)^2 = 1+Sqrt(-3) = 1+i3