Properties

 Label 7T6 Degree $7$ Order $2520$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_7$

Related objects

Group action invariants

 Degree $n$: $7$ Transitive number $t$: $6$ Group: $A_7$ CHM label: $A7$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (3,4,5,6,7), (1,2,3)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

15T47 x 2, 21T33, 35T28, 42T294, 42T299

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 $1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 1, 1, 1$ $105$ $2$ $(1,4)(2,7)$ $4, 2, 1$ $630$ $4$ $(1,2,4,7)(3,5)$ $5, 1, 1$ $504$ $5$ $(2,7,6,4,3)$ $7$ $360$ $7$ $(1,3,7,2,6,4,5)$ $7$ $360$ $7$ $(1,5,4,6,2,7,3)$ $3, 3, 1$ $280$ $3$ $(2,7,5)(3,6,4)$ $3, 1, 1, 1, 1$ $70$ $3$ $(3,4,6)$ $3, 2, 2$ $210$ $6$ $(1,5)(2,7)(3,4,6)$

Group invariants

 Order: $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: not available
 Character table:  2 3 . . 2 3 2 2 . . 3 2 . . 2 1 1 . 2 . 5 1 . . . . . . . 1 7 1 1 1 . . . . . . 1a 7a 7b 3a 2a 6a 4a 3b 5a 2P 1a 7a 7b 3a 1a 3a 2a 3b 5a 3P 1a 7b 7a 1a 2a 2a 4a 1a 5a 5P 1a 7b 7a 3a 2a 6a 4a 3b 1a 7P 1a 1a 1a 3a 2a 6a 4a 3b 5a X.1 1 1 1 1 1 1 1 1 1 X.2 6 -1 -1 3 2 -1 . . 1 X.3 10 A /A 1 -2 1 . 1 . X.4 10 /A A 1 -2 1 . 1 . X.5 14 . . 2 2 2 . -1 -1 X.6 14 . . -1 2 -1 . 2 -1 X.7 15 1 1 3 -1 -1 -1 . . X.8 21 . . -3 1 1 -1 . 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