# Properties

 Label 21T33 Order $$2520$$ n $$21$$ 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) 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) $|\Aut(F/K)|$: $1$

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

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