# Properties

 Label 21T13 Degree $21$ Order $147$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_7\times C_7:C_3$

# Related objects

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$7$:  $C_7$
$21$:  $C_7:C_3$, $C_{21}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 7: None

## Low degree siblings

21T13

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

## Group invariants

 Order: $147=3 \cdot 7^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [147, 3]
 Character table: not available.