# Properties

 Label 21T33 Degree $21$ Order $2520$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_7$

# Related objects

## Group action invariants

 Degree $n$: $21$ Transitive number $t$: $33$ Group: $A_7$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,7,12,16,19,21,6)(2,8,13,17,20,5,11)(3,9,14,18,4,10,15), (4,6,5)(9,11,10)(13,15,14)(16,18,17)(19,20,21)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 3: None

Degree 7: None

## Low degree siblings

7T6, 15T47 x 2, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 7, 7$ $360$ $7$ $( 1, 8,12,13,19,21, 6)( 2, 9,17,15, 4,10,18)( 3, 7,16,14,20, 5,11)$ $7, 7, 7$ $360$ $7$ $( 1, 6,21,19,13,12, 8)( 2,18,10, 4,15,17, 9)( 3,11, 5,20,14,16, 7)$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $105$ $2$ $( 1, 7)( 3,15)( 4,13)( 5,14)( 6,12)( 8,11)(16,20)(17,21)$ $3, 3, 3, 3, 3, 3, 3$ $280$ $3$ $( 1,19,15)( 2,10,20)( 3,17,18)( 4,14,11)( 5,21, 6)( 7, 9,13)( 8,16,12)$ $3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1$ $70$ $3$ $( 1, 2, 4)( 7,13, 9)( 8,12,16)(10,14,19)(11,15,20)$ $6, 6, 3, 2, 2, 1, 1$ $210$ $6$ $( 1,19, 2,10, 4,14)( 3,21)( 6,17)( 7, 9,13)( 8,20,12,11,16,15)$ $5, 5, 5, 5, 1$ $504$ $5$ $( 2, 3, 5, 4, 6)( 7, 8,10, 9,11)(12,17,19,20,15)(13,18,14,16,21)$ $4, 4, 4, 4, 2, 2, 1$ $630$ $4$ $( 1, 3, 4, 6)( 2, 5)( 7,17,13,21)( 8,16,20,11)( 9,18)(10,12,19,15)$

## 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 . . 3 2 2 2 . . 3 2 . . 1 . 2 1 2 . 5 1 . . . . . . . 1 7 1 1 1 . . . . . . 1a 7a 7b 2a 4a 3a 6a 3b 5a 2P 1a 7a 7b 1a 2a 3a 3a 3b 5a 3P 1a 7b 7a 2a 4a 1a 2a 1a 5a 5P 1a 7b 7a 2a 4a 3a 6a 3b 1a 7P 1a 1a 1a 2a 4a 3a 6a 3b 5a X.1 1 1 1 1 1 1 1 1 1 X.2 6 -1 -1 2 . 3 -1 . 1 X.3 10 A /A -2 . 1 1 1 . X.4 10 /A A -2 . 1 1 1 . X.5 14 . . 2 . 2 2 -1 -1 X.6 14 . . 2 . -1 -1 2 -1 X.7 15 1 1 -1 -1 3 -1 . . X.8 21 . . 1 -1 -3 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