# Properties

 Label 18T48 Degree $18$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_6^2:C_3$

# Related objects

## Group action invariants

 Degree $n$: $18$ Transitive number $t$: $48$ Group: $C_6^2:C_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (1,9,18)(2,10,17)(3,12,13)(4,11,14)(5,7,16)(6,8,15), (1,13,10)(2,14,9)(3,16,11)(4,15,12)(5,18,8)(6,17,7)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$12$:  $A_4$
$27$:  $C_3^2:C_3$
$36$:  $C_3\times A_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 9: $C_3^2:C_3$

## Low degree siblings

18T48 x 2

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

## Group invariants

 Order: $108=2^{2} \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [108, 22]
 Character table:  2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . . . . . . 3 3 2 2 2 2 2 2 2 2 2 3 2 2 3 2 2 2 2 2 2 1a 2a 3a 6a 6b 3b 6c 6d 6e 6f 3c 6g 6h 3d 3e 3f 3g 3h 3i 3j 2P 1a 1a 3b 3b 3a 3a 3b 3b 3a 3a 3d 3d 3c 3c 3h 3j 3i 3e 3g 3f 3P 1a 2a 1a 2a 2a 1a 2a 2a 2a 2a 1a 2a 2a 1a 1a 1a 1a 1a 1a 1a 5P 1a 2a 3b 6b 6a 3a 6f 6e 6d 6c 3d 6h 6g 3c 3h 3j 3i 3e 3g 3f X.1 1 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 A A A /A /A /A X.3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 /A /A /A A A A X.4 1 1 A A /A /A A A /A /A 1 1 1 1 1 A /A 1 A /A X.5 1 1 /A /A A A /A /A A A 1 1 1 1 1 /A A 1 /A A X.6 1 1 A A /A /A A A /A /A 1 1 1 1 A /A 1 /A 1 A X.7 1 1 /A /A A A /A /A A A 1 1 1 1 /A A 1 A 1 /A X.8 1 1 A A /A /A A A /A /A 1 1 1 1 /A 1 A A /A 1 X.9 1 1 /A /A A A /A /A A A 1 1 1 1 A 1 /A /A A 1 X.10 3 -1 3 -1 -1 3 -1 -1 -1 -1 3 -1 -1 3 . . . . . . X.11 3 3 . . . . . . . . /B /B B B . . . . . . X.12 3 3 . . . . . . . . B B /B /B . . . . . . X.13 3 -1 B -/A -A /B -/A -/A -A -A 3 -1 -1 3 . . . . . . X.14 3 -1 /B -A -/A B -A -A -/A -/A 3 -1 -1 3 . . . . . . X.15 3 -1 . C /C . 2 /C C 2 /B -A -/A B . . . . . . X.16 3 -1 . /C C . 2 C /C 2 B -/A -A /B . . . . . . X.17 3 -1 . 2 2 . /C C /C C /B -A -/A B . . . . . . X.18 3 -1 . 2 2 . C /C C /C B -/A -A /B . . . . . . X.19 3 -1 . /C C . C 2 2 /C /B -A -/A B . . . . . . X.20 3 -1 . C /C . /C 2 2 C B -/A -A /B . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 3*E(3) = (-3+3*Sqrt(-3))/2 = 3b3 C = 2*E(3) = -1+Sqrt(-3) = 2b3