# Properties

 Label 21T21 Degree $21$ Order $441$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7^2:C_3:C_3$

# Related objects

## Group action invariants

 Degree $n$: $21$ Transitive number $t$: $21$ Group: $C_7^2:C_3:C_3$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,2,4)(3,6,5)(8,12,13)(9,14,10)(15,18,17)(16,20,21), (1,8,21,7,10,17,6,12,20,5,14,16,4,9,19,3,11,15,2,13,18)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$21$:  $C_7:C_3$ x 2
$63$:  21T7 x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 7: None

## Low degree siblings

21T21

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $9$ $7$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ $7, 7, 1, 1, 1, 1, 1, 1, 1$ $9$ $7$ $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ $3, 3, 3, 3, 3, 3, 1, 1, 1$ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ $3, 3, 3, 3, 3, 3, 1, 1, 1$ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ $7, 7, 7$ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$ $7, 7, 7$ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ $7, 7, 7$ $9$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ $7, 7, 7$ $3$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ $7, 7, 7$ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ $7, 7, 7$ $3$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 8,15)( 2,11,17)( 3,14,19)( 4,10,21)( 5,13,16)( 6, 9,18)( 7,12,20)$ $21$ $21$ $21$ $( 1, 8,16, 4,10,15, 7,12,21, 3,14,20, 6, 9,19, 2,11,18, 5,13,17)$ $21$ $21$ $21$ $( 1, 8,18, 3,14,15, 5,13,19, 7,12,16, 2,11,20, 4,10,17, 6, 9,21)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1, 8,15)( 2,13,19)( 3,11,16)( 4, 9,20)( 5,14,17)( 6,12,21)( 7,10,18)$ $21$ $21$ $21$ $( 1, 8,16, 2,13,20, 3,11,17, 4, 9,21, 5,14,18, 6,12,15, 7,10,19)$ $21$ $21$ $21$ $( 1, 8,18, 4, 9,16, 7,10,21, 3,11,19, 6,12,17, 2,13,15, 5,14,20)$ $3, 3, 3, 3, 3, 3, 3$ $49$ $3$ $( 1, 8,15)( 2,14,16)( 3,13,17)( 4,12,18)( 5,11,19)( 6,10,20)( 7, 9,21)$ $3, 3, 3, 3, 3, 3, 3$ $49$ $3$ $( 1,15, 8)( 2,16,14)( 3,17,13)( 4,18,12)( 5,19,11)( 6,20,10)( 7,21, 9)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1,15, 8)( 2,17,11)( 3,19,14)( 4,21,10)( 5,16,13)( 6,18, 9)( 7,20,12)$ $21$ $21$ $21$ $( 1,15, 9, 7,20,13, 6,18,10, 5,16,14, 4,21,11, 3,19, 8, 2,17,12)$ $21$ $21$ $21$ $( 1,15,11, 5,16, 9, 2,17,14, 6,18,12, 3,19,10, 7,20, 8, 4,21,13)$ $3, 3, 3, 3, 3, 3, 3$ $7$ $3$ $( 1,15, 8)( 2,19,13)( 3,16,11)( 4,20, 9)( 5,17,14)( 6,21,12)( 7,18,10)$ $21$ $21$ $21$ $( 1,15, 9, 5,17, 8, 2,19,14, 6,21,13, 3,16,12, 7,18,11, 4,20,10)$ $21$ $21$ $21$ $( 1,15,11, 6,21, 8, 4,20,12, 2,19, 9, 7,18,13, 5,17,10, 3,16,14)$

## Group invariants

 Order: $441=3^{2} \cdot 7^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [441, 9]
 Character table: not available.