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Properties

Label 21T47
Degree $21$
Order $8232$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:S_4$

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Show commands: Magma

magma: G := TransitiveGroup(21, 47);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $47$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^3:S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15,6,18)(2,17,5,16)(3,19,4,21)(7,20)(8,14,13,12,11,10,9), (1,9)(2,13,7,12)(3,10,6,8)(4,14,5,11)(15,17,19,21,16,18,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$24$:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

28T348, 42T541, 42T542, 42T543, 42T547

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,11,14,10,13, 9,12)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,10,12,14, 9,11,13)$
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,14,13,12,11,10, 9)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $24$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $24$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $24$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $12$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)$
$ 7, 7, 7 $ $8$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$ 7, 7, 7 $ $8$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $24$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $8$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $147$ $2$ $( 2, 7)( 3, 6)( 4, 5)(16,21)(17,20)(18,19)$
$ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ $294$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,11,14,10,13, 9,12)(16,21)(17,20)(18,19)$
$ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ $294$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,10,12,14, 9,11,13)(16,21)(17,20)(18,19)$
$ 7, 2, 2, 2, 2, 2, 2, 1, 1 $ $294$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,14,13,12,11,10, 9)(16,21)(17,20)(18,19)$
$ 21 $ $392$ $21$ $( 1,11,16, 7,14,18, 6,10,20, 5,13,15, 4, 9,17, 3,12,19, 2, 8,21)$
$ 21 $ $392$ $21$ $( 1,14,18, 6,13,15, 4,12,19, 2,11,16, 7,10,20, 5, 9,17, 3, 8,21)$
$ 21 $ $392$ $21$ $( 1,13,15, 4,11,16, 7, 9,17, 3,14,18, 6,12,19, 2,10,20, 5, 8,21)$
$ 21 $ $392$ $21$ $( 1,10,20, 5,12,19, 2,14,18, 6, 9,17, 3,11,16, 7,13,15, 4, 8,21)$
$ 3, 3, 3, 3, 3, 3, 3 $ $392$ $3$ $( 1, 8,21)( 2,12,19)( 3, 9,17)( 4,13,15)( 5,10,20)( 6,14,18)( 7,11,16)$
$ 21 $ $392$ $21$ $( 1, 9,17, 3,10,20, 5,11,16, 7,12,19, 2,13,15, 4,14,18, 6, 8,21)$
$ 21 $ $392$ $21$ $( 1,12,19, 2, 9,17, 3,13,15, 4,10,20, 5,14,18, 6,11,16, 7, 8,21)$
$ 4, 4, 4, 2, 1, 1, 1, 1, 1, 1, 1 $ $294$ $4$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,11,14,10,13, 9,12)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,10,12,14, 9,11,13)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,14,13,12,11,10, 9)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,12, 9,13,10,14,11)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8,13,11, 9,14,12,10)$
$ 7, 4, 4, 4, 2 $ $294$ $28$ $( 1,15, 6,18)( 2,17, 5,16)( 3,19, 4,21)( 7,20)( 8, 9,10,11,12,13,14)$
$ 14, 2, 2, 2, 1 $ $588$ $14$ $( 1,15, 6,19, 4,16, 2,20, 7,17, 5,21, 3,18)( 9,14)(10,13)(11,12)$
$ 14, 2, 2, 2, 1 $ $588$ $14$ $( 1,15, 7,17, 6,19, 5,21, 4,16, 3,18, 2,20)( 9,14)(10,13)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $294$ $2$ $( 1,15)( 2,20)( 3,18)( 4,16)( 5,21)( 6,19)( 7,17)( 9,14)(10,13)(11,12)$
$ 14, 2, 2, 2, 1 $ $588$ $14$ $( 1,15, 5,21, 2,20, 6,19, 3,18, 7,17, 4,16)( 9,14)(10,13)(11,12)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $8232=2^{3} \cdot 3 \cdot 7^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  8232.bu
magma: IdentifyGroup(G);
 
Character table:    43 x 43 character table

magma: CharacterTable(G);