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Properties

Label 21T29
Degree $21$
Order $1764$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7^2:(C_6\times S_3)$

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Show commands: Magma

magma: G := TransitiveGroup(21, 29);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7^2:(C_6\times S_3)$
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:  (1,4,3)(2,6,7)(8,15,11,16,9,20)(10,18,12,21,13,19)(14,17), (1,15,5,17,6,21)(2,19,7,18,3,16)(4,20)(8,12,14)(10,13,11), (1,4,6,5,2,7)(8,19,10,20,14,15)(9,16,12,21,11,17)(13,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 7: None

Low degree siblings

14T37, 21T29, 28T170, 42T223 x 2, 42T224 x 2, 42T225 x 2, 42T252, 42T253, 42T254, 42T255

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{21}$ $1$ $1$ $()$
$7^{3}$ $18$ $7$ $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$
$7^{2},1^{7}$ $18$ $7$ $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$
$7^{3}$ $12$ $7$ $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$
$3^{6},1^{3}$ $49$ $3$ $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$
$3^{6},1^{3}$ $49$ $3$ $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$
$3^{7}$ $98$ $3$ $( 1,14,21)( 2,13,15)( 3,12,16)( 4,11,17)( 5,10,18)( 6, 9,19)( 7, 8,20)$
$21$ $84$ $21$ $( 1,11,16, 2,14,18, 3,10,20, 4,13,15, 5, 9,17, 6,12,19, 7, 8,21)$
$3^{7}$ $14$ $3$ $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$
$21$ $84$ $21$ $( 1,13,15, 3, 9,16, 5,12,17, 7, 8,18, 2,11,19, 4,14,20, 6,10,21)$
$3^{7}$ $14$ $3$ $( 1,10,17)( 2, 8,21)( 3,13,18)( 4,11,15)( 5, 9,19)( 6,14,16)( 7,12,20)$
$6^{2},3^{2},2,1$ $147$ $6$ $( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$
$2^{7},1^{7}$ $21$ $2$ $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$
$14,7$ $126$ $14$ $( 1, 7, 6, 5, 4, 3, 2)( 8,18,14,19,13,20,12,21,11,15,10,16, 9,17)$
$6^{2},3^{2},2,1$ $147$ $6$ $( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$
$2^{9},1^{3}$ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$6^{3},1^{3}$ $49$ $6$ $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$
$6^{3},1^{3}$ $49$ $6$ $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$
$6^{3},3$ $98$ $6$ $( 1,14,19, 5,11,16)( 2, 8,20, 4,10,15)( 3, 9,21)( 6,12,17, 7,13,18)$
$6^{3},3$ $98$ $6$ $( 1,11,19, 3,12,16)( 2, 8,21)( 4, 9,18, 7,14,17)( 5,13,20, 6,10,15)$
$6^{3},3$ $98$ $6$ $( 1,13,21, 5,14,16)( 2, 8,18, 4,12,19)( 3,10,15)( 6, 9,20, 7,11,17)$
$6^{3},2,1$ $147$ $6$ $( 2, 6, 5, 7, 3, 4)( 8,20, 9,15,13,16)(10,17)(11,19,14,18,12,21)$
$14,2^{3},1$ $126$ $14$ $( 2, 7)( 3, 6)( 4, 5)( 8,21,13,19,11,17, 9,15,14,20,12,18,10,16)$
$2^{10},1$ $21$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)$
$6^{3},2,1$ $147$ $6$ $( 2, 4, 3, 7, 5, 6)( 8,18, 9,15,11,16)(10,19,13,17,12,20)(14,21)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1764.134
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 7A 7B 7C 14A 14B 21A1 21A-1
Size 1 21 21 49 14 14 49 49 98 49 49 98 98 98 147 147 147 147 12 18 18 126 126 84 84
2 P 1A 1A 1A 1A 3A-1 3A1 3B-1 3B1 3C 3B1 3B-1 3C 3A1 3A-1 3B1 3B-1 3B-1 3B1 7A 7B 7C 7B 7C 21A-1 21A1
3 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 2C 2C 2C 2C 2C 2A 2A 2B 2B 7A 7B 7C 14A 14B 7A 7A
7 P 1A 2A 2B 2C 3A1 3A-1 3B1 3B-1 3C 6A1 6A-1 6B 6C1 6C-1 6D1 6D-1 6E1 6E-1 1A 1A 1A 2A 2B 3A1 3A-1
Type
1764.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1764.134.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 1
1764.134.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 −1 1 1 1
1764.134.1d R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1 −1 1 1
1764.134.1e1 C 1 1 1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 ζ3−1 ζ3
1764.134.1e2 C 1 1 1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 ζ3 ζ3−1
1764.134.1f1 C 1 −1 −1 1 ζ3−1 ζ3 ζ3−1 ζ3 1 ζ3 ζ3−1 1 ζ3 ζ3−1 −ζ3 −ζ3−1 −ζ3−1 −ζ3 1 1 1 −1 −1 ζ3−1 ζ3
1764.134.1f2 C 1 −1 −1 1 ζ3 ζ3−1 ζ3 ζ3−1 1 ζ3−1 ζ3 1 ζ3−1 ζ3 −ζ3−1 −ζ3 −ζ3 −ζ3−1 1 1 1 −1 −1 ζ3 ζ3−1
1764.134.1g1 C 1 −1 1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 −ζ3 −ζ3−1 ζ3−1 ζ3 1 1 1 −1 1 ζ3−1 ζ3
1764.134.1g2 C 1 −1 1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 −ζ3−1 −ζ3 ζ3 ζ3−1 1 1 1 −1 1 ζ3 ζ3−1
1764.134.1h1 C 1 1 −1 −1 ζ3−1 ζ3 ζ3−1 ζ3 1 −ζ3 −ζ3−1 −1 −ζ3 −ζ3−1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 −1 ζ3−1 ζ3
1764.134.1h2 C 1 1 −1 −1 ζ3 ζ3−1 ζ3 ζ3−1 1 −ζ3−1 −ζ3 −1 −ζ3−1 −ζ3 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 −1 ζ3 ζ3−1
1764.134.2a R 2 0 0 2 −1 −1 2 2 −1 2 2 −1 −1 −1 0 0 0 0 2 2 2 0 0 −1 −1
1764.134.2b R 2 0 0 −2 −1 −1 2 2 −1 −2 −2 1 1 1 0 0 0 0 2 2 2 0 0 −1 −1
1764.134.2c1 C 2 0 0 2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 2ζ3−1 2ζ3 −1 −ζ3−1 −ζ3 0 0 0 0 2 2 2 0 0 −ζ3 −ζ3−1
1764.134.2c2 C 2 0 0 2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 2ζ3 2ζ3−1 −1 −ζ3 −ζ3−1 0 0 0 0 2 2 2 0 0 −ζ3−1 −ζ3
1764.134.2d1 C 2 0 0 −2 −ζ3 −ζ3−1 2ζ3 2ζ3−1 −1 −2ζ3−1 −2ζ3 1 ζ3−1 ζ3 0 0 0 0 2 2 2 0 0 −ζ3 −ζ3−1
1764.134.2d2 C 2 0 0 −2 −ζ3−1 −ζ3 2ζ3−1 2ζ3 −1 −2ζ3 −2ζ3−1 1 ζ3 ζ3−1 0 0 0 0 2 2 2 0 0 −ζ3−1 −ζ3
1764.134.12a R 12 0 0 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −1 −1
1764.134.12b1 C 12 0 0 0 6ζ3−1 6ζ3 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −ζ3−1 −ζ3
1764.134.12b2 C 12 0 0 0 6ζ3 6ζ3−1 0 0 0 0 0 0 0 0 0 0 0 0 5 −2 −2 0 0 −ζ3 −ζ3−1
1764.134.18a R 18 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 4 −3 0 −1 0 0
1764.134.18b R 18 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 4 −1 0 0 0
1764.134.18c R 18 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 −3 4 1 0 0 0
1764.134.18d R 18 0 −6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −3 4 −3 0 1 0 0

magma: CharacterTable(G);