# Properties

 Label 18T49 Degree $18$ Order $108$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:S_3.C_2$

# Related objects

## Group action invariants

 Degree $n$: $18$ Transitive number $t$: $49$ Group: $C_3^2:S_3.C_2$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $3$ Generators: (1,8,14)(2,9,15)(3,7,13)(4,10,18)(5,11,16)(6,12,17), (1,6,15,10,3,5,14,12,2,4,13,11)(7,17,8,18,9,16)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$36$:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

## Low degree siblings

18T49

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

## Group invariants

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