# Properties

 Label 36T49 Order $$72$$ n $$36$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $F_9$

## Group action invariants

 Degree $n$ : $36$ Transitive number $t$ : $49$ Group : $F_9$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,3,2,4)(5,13,18,29,34,25,22,10)(6,14,17,30,33,26,21,9)(7,16,20,31,36,28,24,12)(8,15,19,32,35,27,23,11), (1,22,27,6)(2,21,28,5)(3,24,25,8)(4,23,26,7)(9,35,19,29)(10,36,20,30)(11,34,18,31)(12,33,17,32)(13,14)(15,16) $|\Aut(F/K)|$: $4$

## Low degree resolvents

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

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: None

Degree 9: $C_3^2:C_8$

Degree 12: 12T46

Degree 18: 18T28

## Low degree siblings

9T15

Siblings are shown with degree $\leq 10$

Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $9$ $2$ $( 5,34)( 6,33)( 7,36)( 8,35)( 9,30)(10,29)(11,32)(12,31)(13,25)(14,26)(15,27) (16,28)(17,21)(18,22)(19,23)(20,24)$ $4, 4, 4, 4, 4, 4, 4, 4, 2, 2$ $9$ $4$ $( 1, 2)( 3, 4)( 5,18,34,22)( 6,17,33,21)( 7,20,36,24)( 8,19,35,23) ( 9,14,30,26)(10,13,29,25)(11,15,32,27)(12,16,31,28)$ $4, 4, 4, 4, 4, 4, 4, 4, 2, 2$ $9$ $4$ $( 1, 2)( 3, 4)( 5,22,34,18)( 6,21,33,17)( 7,24,36,20)( 8,23,35,19) ( 9,26,30,14)(10,25,29,13)(11,27,32,15)(12,28,31,16)$ $8, 8, 8, 8, 4$ $9$ $8$ $( 1, 3, 2, 4)( 5,13,18,29,34,25,22,10)( 6,14,17,30,33,26,21, 9) ( 7,16,20,31,36,28,24,12)( 8,15,19,32,35,27,23,11)$ $8, 8, 8, 8, 4$ $9$ $8$ $( 1, 3, 2, 4)( 5,25,18,10,34,13,22,29)( 6,26,17, 9,33,14,21,30) ( 7,28,20,12,36,16,24,31)( 8,27,19,11,35,15,23,32)$ $8, 8, 8, 8, 4$ $9$ $8$ $( 1, 4, 2, 3)( 5,10,22,25,34,29,18,13)( 6, 9,21,26,33,30,17,14) ( 7,12,24,28,36,31,20,16)( 8,11,23,27,35,32,19,15)$ $8, 8, 8, 8, 4$ $9$ $8$ $( 1, 4, 2, 3)( 5,29,22,13,34,10,18,25)( 6,30,21,14,33, 9,17,26) ( 7,31,24,16,36,12,20,28)( 8,32,23,15,35,11,19,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1, 5,34)( 2, 6,33)( 3, 7,36)( 4, 8,35)( 9,13,19)(10,14,20)(11,16,18) (12,15,17)(21,27,31)(22,28,32)(23,25,30)(24,26,29)$

## 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 2a 4a 4b 8a 8b 8c 8d 3a 2P 1a 1a 2a 2a 4a 4a 4b 4b 3a 3P 1a 2a 4b 4a 8d 8c 8b 8a 1a 5P 1a 2a 4a 4b 8b 8a 8d 8c 3a 7P 1a 2a 4b 4a 8c 8d 8a 8b 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 A -A B -B /B -/B 1 X.4 1 -1 A -A -B B -/B /B 1 X.5 1 -1 -A A -/B /B -B B 1 X.6 1 -1 -A A /B -/B B -B 1 X.7 1 1 -1 -1 A A -A -A 1 X.8 1 1 -1 -1 -A -A A A 1 X.9 8 . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8)^3