# Properties

 Label 18T31 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $S_3\times A_4$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $31$ Group : $S_3\times A_4$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,9,5,15,8)(2,14,10,6,16,7)(3,17,11)(4,18,12), (1,17,7)(2,18,8)(3,13,9)(4,14,10)(5,16,11)(6,15,12) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
12:  $A_4$
18:  $S_3\times C_3$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 6: $A_4$

Degree 9: $S_3\times C_3$

## Low degree siblings

12T43, 18T32

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 44]
 Character table:  2 3 3 3 3 2 2 . 1 1 . 1 1 3 2 1 1 . 1 2 2 1 2 2 1 2 1a 2a 2b 2c 6a 3a 3b 6b 3c 3d 6c 3e 2P 1a 1a 1a 1a 3a 3a 3d 3e 3e 3b 3c 3c 3P 1a 2a 2b 2c 2a 1a 1a 2b 1a 1a 2b 1a 5P 1a 2a 2b 2c 6a 3a 3d 6c 3e 3b 6b 3c X.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 X.3 1 1 -1 -1 1 1 A -A A /A -/A /A X.4 1 1 -1 -1 1 1 /A -/A /A A -A A X.5 1 1 1 1 1 1 A A A /A /A /A X.6 1 1 1 1 1 1 /A /A /A A A A X.7 2 2 . . -1 -1 -1 . 2 -1 . 2 X.8 2 2 . . -1 -1 -A . B -/A . /B X.9 2 2 . . -1 -1 -/A . /B -A . B X.10 3 -1 -3 1 -1 3 . . . . . . X.11 3 -1 3 -1 -1 3 . . . . . . X.12 6 -2 . . 1 -3 . . . . . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3