# Properties

 Label 18T27 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_2\times C_3:S_3.C_2$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $27$ Group : $C_2\times C_3:S_3.C_2$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,18,6)(2,14,17,5)(3,4)(7,9,11,15)(8,10,12,16), (1,15,9,14)(2,16,10,13)(3,17,7,6)(4,18,8,5) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $C_4\times C_2$
36:  $C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:C_4$

## Low degree siblings

12T40 x 2, 12T41 x 2, 18T27

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 45]
 Character table:  2 3 3 3 3 3 3 3 3 1 1 1 1 3 2 . . . 2 . . . 2 2 2 2 1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b 2P 1a 2a 2a 1a 1a 2a 2a 1a 3a 3a 3b 3b 3P 1a 4b 4a 2a 2b 4d 4c 2c 2b 1a 1a 2b 5P 1a 4a 4b 2a 2b 4c 4d 2c 6a 3a 3b 6b 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 -1 1 1 1 1 1 X.4 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 X.5 1 A -A -1 -1 -A A 1 -1 1 1 -1 X.6 1 -A A -1 -1 A -A 1 -1 1 1 -1 X.7 1 A -A -1 1 A -A -1 1 1 1 1 X.8 1 -A A -1 1 -A A -1 1 1 1 1 X.9 4 . . . -4 . . . -1 1 -2 2 X.10 4 . . . -4 . . . 2 -2 1 -1 X.11 4 . . . 4 . . . -2 -2 1 1 X.12 4 . . . 4 . . . 1 1 -2 -2 A = -E(4) = -Sqrt(-1) = -i