# Properties

 Label 36T555 Order $$360$$ n $$36$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No Group: $A_6$

## Group action invariants

 Degree $n$ : $36$ Transitive number $t$ : $555$ Group : $A_6$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,13,36)(2,16,33)(3,17,34)(4,15,31)(5,18,32)(6,14,35)(7,26,19)(8,29,23)(9,30,20)(10,28,21)(11,25,22)(12,27,24), (1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,34,22)(14,35,21)(15,36,24)(16,31,20)(17,32,19)(18,33,23)(25,30,29)(26,28,27) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: $A_6$ x 2

Degree 9: None

Degree 12: None

Degree 18: None

## Low degree siblings

6T15 x 2, 10T26

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $45$ $2$ $( 3, 5)( 4, 6)( 7,36)( 8,32)( 9,35)(10,33)(11,34)(12,31)(13,28)(14,27)(15,30) (16,26)(17,25)(18,29)(19,21)(22,23)$ $5, 5, 5, 5, 5, 5, 5, 1$ $72$ $5$ $( 2, 3, 6, 4, 5)( 7,29,13,31,22)( 8,30,15,32,20)( 9,26,16,35,24) (10,25,14,34,19)(11,27,17,33,21)(12,28,18,36,23)$ $5, 5, 5, 5, 5, 5, 5, 1$ $72$ $5$ $( 2, 4, 3, 5, 6)( 7,31,29,22,13)( 8,32,30,20,15)( 9,35,26,24,16) (10,34,25,19,14)(11,33,27,21,17)(12,36,28,23,18)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $40$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7,23,16)( 8,24,14)( 9,20,17)(10,19,18)(11,21,13) (12,22,15)(25,28,30)(26,27,29)(31,33,32)(34,35,36)$ $4, 4, 4, 4, 4, 4, 4, 4, 2, 2$ $90$ $4$ $( 1, 7,17,31)( 2, 8,15,35)( 3,10,16,36)( 4, 9,14,34)( 5,12,18,33)( 6,11,13,32) (19,25)(20,29,21,26)(22,30)(23,28,24,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $40$ $3$ $( 1, 7,28)( 2,10,26)( 3, 8,29)( 4, 9,27)( 5,11,25)( 6,12,30)(13,33,19) (14,31,24)(15,35,20)(16,36,21)(17,34,23)(18,32,22)$

## Group invariants

 Order: $360=2^{3} \cdot 3^{2} \cdot 5$ Cyclic: No Abelian: No Solvable: No GAP id: [360, 118]
 Character table:  2 3 3 . . . 2 . 3 2 . . . 2 . 2 5 1 . 1 1 . . . 1a 2a 5a 5b 3a 4a 3b 2P 1a 1a 5b 5a 3a 2a 3b 3P 1a 2a 5b 5a 1a 4a 1a 5P 1a 2a 1a 1a 3a 4a 3b X.1 1 1 1 1 1 1 1 X.2 5 1 . . 2 -1 -1 X.3 5 1 . . -1 -1 2 X.4 8 . A *A -1 . -1 X.5 8 . *A A -1 . -1 X.6 9 1 -1 -1 . 1 . X.7 10 -2 . . 1 . 1 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5