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Properties

Label 21T19
Degree $21$
Order $294$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_7:F_7$

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Show commands: Magma

magma: G := TransitiveGroup(21, 19);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:F_7$
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:  (1,7)(2,6)(3,5)(8,13)(9,12)(10,11)(15,18)(16,17)(19,21), (1,14,21,6,8,17)(2,10,16,5,12,15)(3,13,18,4,9,20)(7,11,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$42$:  $F_7$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: None

Low degree siblings

21T19, 42T58 x 2

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, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ $6$ $7$ $( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $49$ $2$ $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$
$ 7, 7, 7 $ $6$ $7$ $( 1, 3, 5, 7, 2, 4, 6)( 8, 9,10,11,12,13,14)(15,21,20,19,18,17,16)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1, 8,15)( 2,11,17, 7,12,20)( 3,14,19, 6, 9,18)( 4,10,21, 5,13,16)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1, 8,15)( 2,12,17)( 3, 9,19)( 4,13,21)( 5,10,16)( 6,14,18)( 7,11,20)$
$ 3, 3, 3, 3, 3, 3, 3 $ $49$ $3$ $( 1,15, 8)( 2,17,12)( 3,19, 9)( 4,21,13)( 5,16,10)( 6,18,14)( 7,20,11)$
$ 6, 6, 6, 3 $ $49$ $6$ $( 1,15, 8)( 2,20,12, 7,17,11)( 3,18, 9, 6,19,14)( 4,16,13, 5,21,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $294=2 \cdot 3 \cdot 7^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  294.14
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A1 3A-1 6A1 6A-1 7A 7B 7C1 7C2 7C3 7D1 7D2 7D3
Size 1 49 49 49 49 49 6 6 6 6 6 6 6 6
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7D2 7D3 7C1 7C2 7D1 7B 7A 7C3
3 P 1A 2A 1A 1A 2A 2A 7D3 7D1 7C2 7C3 7D2 7B 7A 7C1
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A 1A 1A 1A 1A 1A 1A 1A
Type
294.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
294.14.1b R 1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1
294.14.1c1 C 1 1 ζ3−1 ζ3 ζ3 ζ3−1 1 1 1 1 1 1 1 1
294.14.1c2 C 1 1 ζ3 ζ3−1 ζ3−1 ζ3 1 1 1 1 1 1 1 1
294.14.1d1 C 1 −1 ζ3−1 ζ3 −ζ3 −ζ3−1 1 1 1 1 1 1 1 1
294.14.1d2 C 1 −1 ζ3 ζ3−1 −ζ3−1 −ζ3 1 1 1 1 1 1 1 1
294.14.6a R 6 0 0 0 0 0 −1 6 −1 −1 −1 −1 −1 −1
294.14.6b R 6 0 0 0 0 0 6 −1 −1 −1 −1 −1 −1 −1
294.14.6c1 R 6 0 0 0 0 0 −1 −1 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7
294.14.6c2 R 6 0 0 0 0 0 −1 −1 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72
294.14.6c3 R 6 0 0 0 0 0 −1 −1 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73
294.14.6d1 R 6 0 0 0 0 0 −1 −1 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73
294.14.6d2 R 6 0 0 0 0 0 −1 −1 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 2ζ7−1+2+2ζ7 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73
294.14.6d3 R 6 0 0 0 0 0 −1 −1 2ζ7−1+2+2ζ7 2ζ7−2+2+2ζ72 2ζ7−3+2+2ζ73 ζ7−3+2ζ7−2+2ζ72+ζ73 ζ7−3−ζ7−2−1−ζ72+ζ73 −2ζ7−3−ζ7−2−2−ζ72−2ζ73

magma: CharacterTable(G);