# Properties

 Label 15T47 Order $$2520$$ n $$15$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $A_7$

# Related objects

## Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $47$ Group : $A_7$ CHM label : $A_{7}(15)$ 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), (1,2,3)(5,6,7)(8,10,9)(12,14,13) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 5: None

## Low degree siblings

7T6, 15T47, 21T33, 35T28, 42T294, 42T299

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1$ $105$ $2$ $( 1, 5)( 2, 7)( 3, 6)( 4,15)( 8, 9)(12,13)$ $3, 3, 3, 3, 3$ $70$ $3$ $( 1, 2,12)( 3, 8, 4)( 5, 7,13)( 6, 9,15)(10,14,11)$ $3, 3, 3, 3, 1, 1, 1$ $280$ $3$ $( 1, 3, 9)( 2, 8,15)( 4, 6,12)( 5, 7,13)$ $6, 6, 3$ $210$ $6$ $( 1, 7,12, 5, 2,13)( 3, 9, 4, 6, 8,15)(10,14,11)$ $5, 5, 5$ $504$ $5$ $( 1, 8,12,14, 4)( 2, 9, 7,15, 3)( 5,10, 6,11,13)$ $7, 7, 1$ $360$ $7$ $( 1, 3, 6, 8, 4,13,10)( 2, 5,14,12, 9, 7,11)$ $7, 7, 1$ $360$ $7$ $( 1,10,13, 4, 8, 6, 3)( 2,11, 7, 9,12,14, 5)$ $4, 4, 4, 2, 1$ $630$ $4$ $( 1, 6, 5, 3)( 2,15, 7, 4)( 8,12, 9,13)(10,11)$

## Group invariants

 Order: $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 3 3 2 . . . 2 2 . 3 2 1 . . . . 2 1 2 5 1 . . . . 1 . . . 7 1 . . 1 1 . . . . 1a 2a 4a 7a 7b 5a 3a 6a 3b 2P 1a 1a 2a 7a 7b 5a 3a 3a 3b 3P 1a 2a 4a 7b 7a 5a 1a 2a 1a 5P 1a 2a 4a 7b 7a 1a 3a 6a 3b 7P 1a 2a 4a 1a 1a 5a 3a 6a 3b X.1 1 1 1 1 1 1 1 1 1 X.2 6 2 . -1 -1 1 3 -1 . X.3 10 -2 . A /A . 1 1 1 X.4 10 -2 . /A A . 1 1 1 X.5 14 2 . . . -1 2 2 -1 X.6 14 2 . . . -1 -1 -1 2 X.7 15 -1 -1 1 1 . 3 -1 . X.8 21 1 -1 . . 1 -3 1 . X.9 35 -1 1 . . . -1 -1 -1 A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7