# Properties

 Label 18T8 Order $$36$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $A_4 \times C_3$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $8$ Group : $A_4 \times C_3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,16,10)(2,15,9)(3,18,11)(4,17,12)(5,13,8)(6,14,7), (1,8,17)(2,7,18)(3,10,14)(4,9,13)(5,11,15)(6,12,16) $|\Aut(F/K)|$: $6$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$ x 4

Degree 6: $A_4$

Degree 9: $C_3^2$

## Low degree siblings

12T20 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $3$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$ $3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$ $6, 6, 3, 3$ $3$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,18,14,16,17)$ $6, 6, 3, 3$ $3$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,11, 9, 8,12,10)(13,17,16,14,18,15)$ $3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,16)(14,17,15)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,15)( 6,11,16)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1, 9,16)( 2,10,15)( 3,12,18)( 4,11,17)( 5, 7,13)( 6, 8,14)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,11,13)( 2,12,14)( 3, 8,16)( 4, 7,15)( 5,10,18)( 6, 9,17)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,18, 9)( 6,17,10)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,15, 9)( 2,16,10)( 3,17,12)( 4,18,11)( 5,14, 7)( 6,13, 8)$ $3, 3, 3, 3, 3, 3$ $4$ $3$ $( 1,17, 7)( 2,18, 8)( 3,14, 9)( 4,13,10)( 5,15,12)( 6,16,11)$

## Group invariants

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