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Properties

Label 21T44
Degree $21$
Order $7560$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no

Related objects

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Group action invariants

Degree $n$:  $21$
Transitive number $t$:  $44$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $3$
Generators:  (1,11,17,21,7,5,13,2,12,18,19,8,6,14,3,10,16,20,9,4,15), (1,16,15,10,2,17,13,11,3,18,14,12)(4,8,6,7,5,9)(19,20,21)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$
$2520$:  $A_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $A_7$

Low degree siblings

45T442 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 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1, 6)( 2, 4)( 3, 5)( 7,11)( 8,12)( 9,10)$
$ 6, 6, 3, 3, 3 $ $105$ $6$ $( 1, 5, 2, 6, 3, 4)( 7,10, 8,11, 9,12)(13,15,14)(16,18,17)(19,21,20)$
$ 6, 6, 3, 3, 3 $ $105$ $6$ $( 1, 4, 3, 6, 2, 5)( 7,12, 9,11, 8,10)(13,14,15)(16,17,18)(19,20,21)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)$
$ 3, 3, 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ $210$ $6$ $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)$
$ 6, 6, 3, 3, 3 $ $210$ $6$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,15,11,13,12,14)(16,21,17,19,18,20)$
$ 6, 6, 3, 3, 3 $ $210$ $6$ $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,14,12,13,11,15)(16,20,18,19,17,21)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $280$ $3$ $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)(10,13,18)(11,14,16)(12,15,17)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,15,16)(11,13,17)(12,14,18)(19,21,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $280$ $3$ $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,14,17)(11,15,18)(12,13,16)(19,20,21)$
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $630$ $4$ $( 1, 6, 9,10)( 2, 4, 7,11)( 3, 5, 8,12)(13,18)(14,16)(15,17)$
$ 12, 6, 3 $ $630$ $12$ $( 1, 5, 7,10, 3, 4, 9,12, 2, 6, 8,11)(13,17,14,18,15,16)(19,21,20)$
$ 12, 6, 3 $ $630$ $12$ $( 1, 4, 8,10, 2, 5, 9,11, 3, 6, 7,12)(13,16,15,18,14,17)(19,20,21)$
$ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ $504$ $5$ $( 1, 6, 9,10,13)( 2, 4, 7,11,14)( 3, 5, 8,12,15)$
$ 15, 3, 3 $ $504$ $15$ $( 1, 5, 7,10,15, 2, 6, 8,11,13, 3, 4, 9,12,14)(16,18,17)(19,21,20)$
$ 15, 3, 3 $ $504$ $15$ $( 1, 4, 8,10,14, 3, 6, 7,12,13, 2, 5, 9,11,15)(16,17,18)(19,20,21)$
$ 7, 7, 7 $ $360$ $7$ $( 1, 6, 9,10,13,18,21)( 2, 4, 7,11,14,16,19)( 3, 5, 8,12,15,17,20)$
$ 21 $ $360$ $21$ $( 1, 5, 7,10,15,16,21, 3, 4, 9,12,14,18,20, 2, 6, 8,11,13,17,19)$
$ 21 $ $360$ $21$ $( 1, 4, 8,10,14,17,21, 2, 5, 9,11,15,18,19, 3, 6, 7,12,13,16,20)$
$ 7, 7, 7 $ $360$ $7$ $( 1, 6, 9,10,13,21,18)( 2, 4, 7,11,14,19,16)( 3, 5, 8,12,15,20,17)$
$ 21 $ $360$ $21$ $( 1, 5, 7,10,15,19,18, 3, 4, 9,12,14,21,17, 2, 6, 8,11,13,20,16)$
$ 21 $ $360$ $21$ $( 1, 4, 8,10,14,20,18, 2, 5, 9,11,15,21,16, 3, 6, 7,12,13,19,17)$

Group invariants

Order:  $7560=2^{3} \cdot 3^{3} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.