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Properties

Label 21T35
Degree $21$
Order $3087$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^3:C_9$

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Show commands: Magma

magma: G := TransitiveGroup(21, 35);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^3:C_9$
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:  (1,12,16,5,10,21,7,9,20)(2,8,19)(3,11,15,6,13,17,4,14,18), (1,5,2,6,3,7,4)(8,11,14,10,13,9,12)(15,16,17,18,19,20,21)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$3$:  $C_3$
$9$:  $C_9$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T35 x 18

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 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 7, 5)( 3, 4, 6)( 9,10,12)(11,14,13)(15,18,17)(16,20,21)$
3A-1 $3^{6},1^{3}$ $343$ $3$ $12$ $( 1, 5, 7)( 3, 6, 4)( 9,12,10)(11,13,14)(15,17,18)(16,21,20)$
7A1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$
7A-1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$
7B1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
7B-1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
7C1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$
7C-1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$
7D1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
7D-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)$
7E1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$
7E-1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,18,21,17,20,16,19)$
7F1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
7F-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)$
7G1 $7^{3}$ $9$ $7$ $18$ $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7G-1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$
7H1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7H-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 5, 2, 6, 3, 7, 4)(15,17,19,21,16,18,20)$
7I1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$
7I-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 7, 6, 5, 4, 3, 2)(15,20,18,16,21,19,17)$
7J1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 2, 3, 4, 5, 6, 7)(15,21,20,19,18,17,16)$
7J-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 3, 5, 7, 2, 4, 6)(15,18,21,17,20,16,19)$
7K1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,11,14,10,13, 9,12)(15,20,18,16,21,19,17)$
7K-1 $7^{3}$ $9$ $7$ $18$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$
7L1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,12, 9,13,10,14,11)(15,16,17,18,19,20,21)$
7L-1 $7^{3}$ $9$ $7$ $18$ $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$
7M1 $7,1^{14}$ $9$ $7$ $6$ $(1,4,7,3,6,2,5)$
7M-1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
7N1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 8,12, 9,13,10,14,11)(15,17,19,21,16,18,20)$
7N-1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$
7O1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$
7O-1 $7^{3}$ $9$ $7$ $18$ $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
7P1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 6, 4, 2, 7, 5, 3)(15,21,20,19,18,17,16)$
7P-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
7Q1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 1, 6, 4, 2, 7, 5, 3)(15,16,17,18,19,20,21)$
7Q-1 $7^{2},1^{7}$ $9$ $7$ $12$ $( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
7R1 $7,1^{14}$ $9$ $7$ $6$ $(1,3,5,7,2,4,6)$
7R-1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
7S1 $7^{3}$ $9$ $7$ $18$ $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
7S-1 $7^{3}$ $9$ $7$ $18$ $( 1, 3, 5, 7, 2, 4, 6)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$
9A1 $9^{2},3$ $343$ $9$ $18$ $( 1, 9,21, 5,12,20, 7,10,16)( 2, 8,19)( 3,14,17, 6,11,18, 4,13,15)$
9A-1 $9^{2},3$ $343$ $9$ $18$ $( 1,10,20, 5, 9,16, 7,12,21)( 2, 8,19)( 3,13,18, 6,14,15, 4,11,17)$
9A2 $9^{2},3$ $343$ $9$ $18$ $( 1,20, 9, 7,21,10, 5,16,12)( 2,19, 8)( 3,18,14, 4,17,13, 6,15,11)$
9A-2 $9^{2},3$ $343$ $9$ $18$ $( 1,21,12, 7,16, 9, 5,20,10)( 2,19, 8)( 3,17,11, 4,15,14, 6,18,13)$
9A4 $9^{2},3$ $343$ $9$ $18$ $( 1,12,16, 5,10,21, 7, 9,20)( 2, 8,19)( 3,11,15, 6,13,17, 4,14,18)$
9A-4 $9^{2},3$ $343$ $9$ $18$ $( 1,16,10, 7,20,12, 5,21, 9)( 2,19, 8)( 3,15,13, 4,18,11, 6,17,14)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3087=3^{2} \cdot 7^{3}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  3087.f
magma: IdentifyGroup(G);
 
Character table:    47 x 47 character table

magma: CharacterTable(G);