# Properties

 Label 18T24 Degree $18$ Order $54$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3^2:S_3$

# Related objects

## Group action invariants

 Degree $n$: $18$ Transitive number $t$: $24$ Group: $C_3^2:S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $6$ Generators: (5,7,9)(6,8,10)(11,15,13)(12,16,14), (1,8,13)(2,7,14)(3,10,15)(4,9,16)(5,12,18)(6,11,17), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$ x 4
$18$:  $C_3^2:C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 9: $(C_3^2:C_3):C_2$

## Low degree siblings

9T12 x 4, 18T24 x 3

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

## Group invariants

 Order: $54=2 \cdot 3^{3}$ Cyclic: no Abelian: no Solvable: yes GAP id: [54, 8]
 Character table:  2 1 . 1 1 1 1 . . . 1 3 3 2 1 3 1 1 2 2 2 3 1a 3a 2a 3b 6a 6b 3c 3d 3e 3f 2P 1a 3a 1a 3f 3f 3b 3c 3d 3e 3b 3P 1a 1a 2a 1a 2a 2a 1a 1a 1a 1a 5P 1a 3a 2a 3f 6b 6a 3c 3d 3e 3b X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 1 1 1 1 X.3 2 2 . 2 . . -1 -1 -1 2 X.4 2 -1 . 2 . . 2 -1 -1 2 X.5 2 -1 . 2 . . -1 -1 2 2 X.6 2 -1 . 2 . . -1 2 -1 2 X.7 3 . -1 A B /B . . . /A X.8 3 . -1 /A /B B . . . A X.9 3 . 1 A -B -/B . . . /A X.10 3 . 1 /A -/B -B . . . A A = 3*E(3)^2 = (-3-3*Sqrt(-3))/2 = -3-3b3 B = -E(3)^2 = (1+Sqrt(-3))/2 = 1+b3