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Properties

Label 21T48
Degree $21$
Order $9261$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:\He_3$

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Show commands: Magma

magma: G := TransitiveGroup(21, 48);
 

Group invariants

Abstract group:  $C_7^3:\He_3$
magma: IdentifyGroup(G);
 
Order:  $9261=3^{3} \cdot 7^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6,5)(2,3,7)(8,12,13)(9,14,10)$, $(1,15,9,3,20,10,5,18,11,7,16,12,2,21,13,4,19,14,6,17,8)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$27$:  $C_3^2:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T48 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
3A1 $3^{7}$ $147$ $3$ $14$ $( 1,12,18)( 2,13,17)( 3,14,16)( 4, 8,15)( 5, 9,21)( 6,10,20)( 7,11,19)$
3A-1 $3^{7}$ $147$ $3$ $14$ $( 1,10,20)( 2,12,19)( 3,14,18)( 4, 9,17)( 5,11,16)( 6,13,15)( 7, 8,21)$
3B1 $3^{7}$ $147$ $3$ $14$ $( 1,15, 9)( 2,21,13)( 3,20,10)( 4,19,14)( 5,18,11)( 6,17, 8)( 7,16,12)$
3B-1 $3^{7}$ $147$ $3$ $14$ $( 1,20, 8)( 2,18, 9)( 3,16,10)( 4,21,11)( 5,19,12)( 6,17,13)( 7,15,14)$
3C1 $3^{4},1^{9}$ $147$ $3$ $8$ $( 1, 3, 4)( 2, 7, 6)( 8,13, 9)(11,12,14)$
3C-1 $3^{7}$ $147$ $3$ $14$ $( 1,19,13)( 2,15, 8)( 3,18,10)( 4,21,12)( 5,17,14)( 6,20, 9)( 7,16,11)$
3D1 $3^{4},1^{9}$ $147$ $3$ $8$ $( 1, 4, 3)( 2, 6, 7)( 8, 9,13)(11,14,12)$
3D-1 $3^{7}$ $147$ $3$ $14$ $( 1, 9,15)( 2,13,21)( 3,10,20)( 4,14,19)( 5,11,18)( 6, 8,17)( 7,12,16)$
3E1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 5, 6)( 2, 7, 3)( 8,11,10)( 9,13,14)(15,19,20)(16,21,17)$
3E-1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 6, 5)( 2, 3, 7)( 8,10,11)( 9,14,13)(15,20,19)(16,17,21)$
7A1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
7A-1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7B1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7B-1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
7C1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
7C-1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7D1 $7,1^{14}$ $9$ $7$ $6$ $(15,21,20,19,18,17,16)$
7D-1 $7,1^{14}$ $9$ $7$ $6$ $(15,19,16,20,17,21,18)$
7E1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
7E-1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
7F1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7F-1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7G1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$
7G-1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7H1 $7^{2},1^{7}$ $27$ $7$ $12$ $( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$
7H-1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
7I1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7I-1 $7^{3}$ $27$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
21A1 $21$ $441$ $21$ $20$ $( 1,19,12, 5,18, 9, 2,17,13, 6,16,10, 3,15,14, 7,21,11, 4,20, 8)$
21A-1 $21$ $441$ $21$ $20$ $( 1,12,16, 7, 8,17, 6,11,18, 5,14,19, 4,10,20, 3,13,21, 2, 9,15)$
21A2 $21$ $441$ $21$ $20$ $( 1, 8,21, 7,13,15, 6,11,16, 5, 9,17, 4,14,18, 3,12,19, 2,10,20)$
21A-2 $21$ $441$ $21$ $20$ $( 1,11,16, 5,12,19, 2,13,15, 6,14,18, 3, 8,21, 7, 9,17, 4,10,20)$
21B1 $21$ $441$ $21$ $20$ $( 1,18,11, 5,21,13, 2,17, 8, 6,20,10, 3,16,12, 7,19,14, 4,15, 9)$
21B-1 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 1, 4, 3)( 2, 6, 7)( 8, 9,13)(11,14,12)(15,19,16,20,17,21,18)$
21B2 $21$ $441$ $21$ $20$ $( 1,11,19, 7,10,20, 6, 9,21, 5, 8,15, 4,14,16, 3,13,17, 2,12,18)$
21B-2 $21$ $441$ $21$ $20$ $( 1,16,12, 7,17, 8, 6,18,11, 5,19,14, 4,20,10, 3,21,13, 2,15, 9)$
21C1 $21$ $441$ $21$ $20$ $( 1,21,11, 4,15,14, 7,16,10, 3,17,13, 6,18, 9, 2,19,12, 5,20, 8)$
21C-1 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 1, 4, 3)( 2, 6, 7)( 8, 9,13)(11,14,12)(15,18,21,17,20,16,19)$
21C2 $21$ $441$ $21$ $20$ $( 1,14,16, 3, 9,21, 5,11,19, 7,13,17, 2, 8,15, 4,10,20, 6,12,18)$
21C-2 $21$ $441$ $21$ $20$ $( 1,20, 9, 6,21,12, 4,15, 8, 2,16,11, 7,17,14, 5,18,10, 3,19,13)$
21D1 $21$ $441$ $21$ $20$ $( 1,17,14, 5,15, 8, 2,20, 9, 6,18,10, 3,16,11, 7,21,12, 4,19,13)$
21D-1 $21$ $441$ $21$ $20$ $( 1,11,18, 5,13,21, 2, 8,17, 6,10,20, 3,12,16, 7,14,19, 4, 9,15)$
21D2 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 1, 3, 4)( 2, 7, 6)( 8,13, 9)(11,12,14)(15,17,19,21,16,18,20)$
21D-2 $7,3^{4},1^{2}$ $441$ $21$ $14$ $( 1, 3, 4)( 2, 7, 6)( 8,13, 9)(11,12,14)(15,21,20,19,18,17,16)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

45 x 45 character table

magma: CharacterTable(G);
 

Regular extensions

Data not computed