# Properties

 Label 18T28 Order $$72$$ n $$18$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $F_9$

# Related objects

## Group action invariants

 Degree $n$ : $18$ Transitive number $t$ : $28$ Group : $F_9$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,8,14)(2,7,13)(3,9,16)(4,10,15)(5,12,18)(6,11,17), (1,12,3,18,6,13,9,7)(2,11,4,17,5,14,10,8)(15,16) $|\Aut(F/K)|$: $2$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
8:  $C_8$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 6: None

Degree 9: $C_3^2:C_8$

## Low degree siblings

9T15, 12T46

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

## Group invariants

 Order: $72=2^{3} \cdot 3^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [72, 39]
 Character table:  2 3 3 3 3 3 3 3 3 . 3 2 . . . . . . . 2 1a 4a 4b 2a 8a 8b 8c 8d 3a 2P 1a 2a 2a 1a 4b 4a 4a 4b 3a 3P 1a 4b 4a 2a 8c 8d 8a 8b 1a 5P 1a 4a 4b 2a 8d 8c 8b 8a 3a 7P 1a 4b 4a 2a 8b 8a 8d 8c 3a X.1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 -1 -1 -1 -1 1 X.3 1 -1 -1 1 A -A -A A 1 X.4 1 -1 -1 1 -A A A -A 1 X.5 1 A -A -1 B /B -/B -B 1 X.6 1 A -A -1 -B -/B /B B 1 X.7 1 -A A -1 -/B -B B /B 1 X.8 1 -A A -1 /B B -B -/B 1 X.9 8 . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8)