Properties

Label 21T57
Degree $21$
Order $15120$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no

Related objects

Learn more about

Group action invariants

Degree $n$:  $21$
Transitive number $t$:  $57$
Parity:  $-1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,2,3)(4,17,12,19,8,6,16,11,21,7,5,18,10,20,9)(13,14,15), (1,4)(2,6)(3,5)(7,10,19,13)(8,12,20,15)(9,11,21,14)(17,18)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$2520$:  $A_7$
$5040$:  $A_7\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: $A_7$

Low degree siblings

42T667, 45T581 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,10)( 8,11)( 9,12)$
$ 6, 6, 3, 3, 3 $ $210$ $6$ $( 1, 5, 3, 4, 2, 6)( 7,11, 9,10, 8,12)(13,14,15)(16,17,18)(19,20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $315$ $2$ $( 1, 4)( 2, 6)( 3, 5)( 7,10)( 8,12)( 9,11)(14,15)(17,18)(20,21)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)$
$ 3, 3, 3, 3, 3, 3, 3 $ $140$ $3$ $( 1, 5, 9)( 2, 6, 7)( 3, 4, 8)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 6, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $210$ $6$ $( 1, 4, 7)( 2, 6, 8, 3, 5, 9)(11,12)(14,15)(17,18)(20,21)$
$ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ $210$ $6$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)$
$ 6, 6, 3, 3, 3 $ $420$ $6$ $( 1, 5, 9)( 2, 6, 7)( 3, 4, 8)(10,14,12,13,11,15)(16,20,18,19,17,21)$
$ 6, 3, 2, 2, 2, 2, 2, 2 $ $630$ $6$ $( 1, 4, 7)( 2, 6, 8, 3, 5, 9)(10,13)(11,15)(12,14)(16,19)(17,21)(18,20)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $280$ $3$ $( 1, 4, 7)( 2, 5, 8)( 3, 6, 9)(10,13,16)(11,14,17)(12,15,18)$
$ 3, 3, 3, 3, 3, 3, 3 $ $560$ $3$ $( 1, 5, 9)( 2, 6, 7)( 3, 4, 8)(10,14,18)(11,15,16)(12,13,17)(19,20,21)$
$ 6, 6, 3, 3, 2, 1 $ $840$ $6$ $( 1, 4, 7)( 2, 6, 8, 3, 5, 9)(10,13,16)(11,15,17,12,14,18)(20,21)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $630$ $4$ $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)(13,16)(14,17)(15,18)$
$ 12, 6, 3 $ $1260$ $12$ $( 1, 5, 9,10, 2, 6, 7,11, 3, 4, 8,12)(13,17,15,16,14,18)(19,20,21)$
$ 4, 4, 4, 2, 2, 2, 2, 1 $ $1890$ $4$ $( 1, 4, 7,10)( 2, 6, 8,12)( 3, 5, 9,11)(13,16)(14,18)(15,17)(20,21)$
$ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ $504$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15, 3, 3 $ $1008$ $15$ $( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,17,18)(19,20,21)$
$ 10, 5, 2, 2, 1, 1 $ $1512$ $10$ $( 1, 4, 7,10,13)( 2, 6, 8,12,14, 3, 5, 9,11,15)(17,18)(20,21)$
$ 7, 7, 7 $ $360$ $7$ $( 1, 4, 7,10,13,16,19)( 2, 5, 8,11,14,17,20)( 3, 6, 9,12,15,18,21)$
$ 21 $ $720$ $21$ $( 1, 5, 9,10,14,18,19, 2, 6, 7,11,15,16,20, 3, 4, 8,12,13,17,21)$
$ 14, 7 $ $1080$ $14$ $( 1, 4, 7,10,13,16,19)( 2, 6, 8,12,14,18,20, 3, 5, 9,11,15,17,21)$
$ 7, 7, 7 $ $360$ $7$ $( 1, 4, 7,10,13,19,16)( 2, 5, 8,11,14,20,17)( 3, 6, 9,12,15,21,18)$
$ 21 $ $720$ $21$ $( 1, 5, 9,10,14,21,16, 2, 6, 7,11,15,19,17, 3, 4, 8,12,13,20,18)$
$ 14, 7 $ $1080$ $14$ $( 1, 4, 7,10,13,19,16)( 2, 6, 8,12,14,21,17, 3, 5, 9,11,15,20,18)$

Group invariants

Order:  $15120=2^{4} \cdot 3^{3} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.