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Properties

Label 21T9
Degree $21$
Order $126$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times F_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 9);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times F_7$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,13,2,14,3,15)(4,10,5,11,6,12)(7,9,8)(16,19,17,20,18,21), (1,12,8)(2,10,9)(3,11,7)(4,16,20)(5,17,21)(6,18,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$18$:  $C_6 \times C_3$
$42$:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $F_7$

Low degree siblings

21T9 x 2, 42T17 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{21}$ $1$ $1$ $()$
$3^{6},1^{3}$ $7$ $3$ $( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,18)(11,20,16)(12,21,17)$
$6^{3},1^{3}$ $7$ $6$ $( 4,11, 7,20,13,16)( 5,12, 8,21,14,17)( 6,10, 9,19,15,18)$
$3^{6},1^{3}$ $7$ $3$ $( 4,13, 7)( 5,14, 8)( 6,15, 9)(10,18,19)(11,16,20)(12,17,21)$
$6^{3},1^{3}$ $7$ $6$ $( 4,16,13,20, 7,11)( 5,17,14,21, 8,12)( 6,18,15,19, 9,10)$
$2^{9},1^{3}$ $7$ $2$ $( 4,20)( 5,21)( 6,19)( 7,16)( 8,17)( 9,18)(10,15)(11,13)(12,14)$
$3^{7}$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$
$3^{7}$ $7$ $3$ $( 1, 2, 3)( 4, 8,15)( 5, 9,13)( 6, 7,14)(10,20,17)(11,21,18)(12,19,16)$
$6^{3},3$ $7$ $6$ $( 1, 2, 3)( 4,12, 9,20,14,18)( 5,10, 7,21,15,16)( 6,11, 8,19,13,17)$
$3^{7}$ $7$ $3$ $( 1, 2, 3)( 4,14, 9)( 5,15, 7)( 6,13, 8)(10,16,21)(11,17,19)(12,18,20)$
$6^{3},3$ $7$ $6$ $( 1, 2, 3)( 4,17,15,20, 8,10)( 5,18,13,21, 9,11)( 6,16,14,19, 7,12)$
$6^{3},3$ $7$ $6$ $( 1, 2, 3)( 4,21, 6,20, 5,19)( 7,17, 9,16, 8,18)(10,13,12,15,11,14)$
$3^{7}$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$
$3^{7}$ $7$ $3$ $( 1, 3, 2)( 4, 9,14)( 5, 7,15)( 6, 8,13)(10,21,16)(11,19,17)(12,20,18)$
$6^{3},3$ $7$ $6$ $( 1, 3, 2)( 4,10, 8,20,15,17)( 5,11, 9,21,13,18)( 6,12, 7,19,14,16)$
$3^{7}$ $7$ $3$ $( 1, 3, 2)( 4,15, 8)( 5,13, 9)( 6,14, 7)(10,17,20)(11,18,21)(12,16,19)$
$6^{3},3$ $7$ $6$ $( 1, 3, 2)( 4,18,14,20, 9,12)( 5,16,15,21, 7,10)( 6,17,13,19, 8,11)$
$6^{3},3$ $7$ $6$ $( 1, 3, 2)( 4,19, 5,20, 6,21)( 7,18, 8,16, 9,17)(10,14,11,15,12,13)$
$21$ $6$ $21$ $( 1, 4, 9,12,13,18,21, 3, 6, 8,11,15,17,20, 2, 5, 7,10,14,16,19)$
$7^{3}$ $6$ $7$ $( 1, 5, 8,12,14,17,21)( 2, 6, 9,10,15,18,19)( 3, 4, 7,11,13,16,20)$
$21$ $6$ $21$ $( 1, 6, 7,12,15,16,21, 2, 4, 8,10,13,17,19, 3, 5, 9,11,14,18,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $126=2 \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  126.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 3B1 3B-1 3C1 3C-1 3D1 3D-1 6A1 6A-1 6B1 6B-1 6C1 6C-1 6D1 6D-1 7A 21A1 21A-1
Size 1 7 1 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6
2 P 1A 1A 3A-1 3A1 3B1 3B-1 3C-1 3C1 3D1 3D-1 3C1 3B1 3B-1 3D1 3A1 3A-1 3D-1 3C-1 7A 21A-1 21A1
3 P 1A 2A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 7A 7A 7A
7 P 1A 2A 3A1 3A-1 3B-1 3B1 3C1 3C-1 3D-1 3D1 6B1 6A1 6A-1 6D1 6C1 6C-1 6D-1 6B-1 1A 3A1 3A-1
Type
126.7.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
126.7.1b R 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1
126.7.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 ζ3−1 ζ3
126.7.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 ζ3 ζ3−1
126.7.1d1 C 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3
126.7.1d2 C 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1
126.7.1e1 C 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 ζ3−1 ζ3
126.7.1e2 C 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 ζ3 ζ3−1
126.7.1f1 C 1 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 1 1 1
126.7.1f2 C 1 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 ζ3 ζ3−1 1 1 1
126.7.1g1 C 1 −1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1 1 ζ3−1 ζ3
126.7.1g2 C 1 −1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1 1 ζ3 ζ3−1
126.7.1h1 C 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 ζ3−1 ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 1 ζ3−1 ζ3
126.7.1h2 C 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 1 ζ3 ζ3−1
126.7.1i1 C 1 −1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 ζ3−1 ζ3
126.7.1i2 C 1 −1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 ζ3 ζ3−1
126.7.1j1 C 1 −1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 −1 −1 −ζ3−1 −ζ3 1 1 1
126.7.1j2 C 1 −1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 −1 −1 −ζ3 −ζ3−1 1 1 1
126.7.6a R 6 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 −1
126.7.6b1 C 6 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −ζ3−1 −ζ3
126.7.6b2 C 6 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 −ζ3 −ζ3−1

magma: CharacterTable(G);