# Properties

 Label 15T20 Order $$360$$ n $$15$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $A_6$

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## Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $20$ Group : $A_6$ CHM label : $A_{6}(15)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,5)(2,7)(3,6)(4,15)(8,9)(12,13) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 5: None

## Low degree siblings

6T15 x 2, 10T26, 15T20, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 1, 1, 1$ $40$ $3$ $( 3, 8,10)( 4,13, 6)( 5, 7,14)( 9,15,11)$ $2, 2, 2, 2, 2, 2, 1, 1, 1$ $45$ $2$ $( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$ $4, 4, 4, 2, 1$ $90$ $4$ $( 2, 9, 5,15)( 3,10,13, 4)( 6, 8)( 7,11,14,12)$ $5, 5, 5$ $72$ $5$ $( 1, 2, 3, 4,11)( 5,12,14, 8,15)( 6,13, 9, 7,10)$ $5, 5, 5$ $72$ $5$ $( 1, 2, 3, 6, 9)( 4,13,11, 5, 8)( 7,12,14,10,15)$ $3, 3, 3, 3, 3$ $40$ $3$ $( 1, 2,12)( 3, 9, 5)( 4, 6,13)( 7, 8,15)(10,11,14)$

## 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 3a 2a 4a 5a 5b 3b 2P 1a 3a 1a 2a 5b 5a 3b 3P 1a 1a 2a 4a 5b 5a 1a 5P 1a 3a 2a 4a 1a 1a 3b X.1 1 1 1 1 1 1 1 X.2 5 2 1 -1 . . -1 X.3 5 -1 1 -1 . . 2 X.4 8 -1 . . A *A -1 X.5 8 -1 . . *A A -1 X.6 9 . 1 1 -1 -1 . X.7 10 1 -2 . . . 1 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5