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Properties

Label 42T19
Degree $42$
Order $126$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{21}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(42, 19);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{21}:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15,41)(2,16,42)(3,18,37)(4,17,38)(5,14,39)(6,13,40)(7,27,24)(8,28,23)(9,29,20)(10,30,19)(11,26,21)(12,25,22)(31,33,35)(32,34,36), (1,9,13,21,25,33,37,4,8,16,19,27,32,39)(2,10,14,22,26,34,38,3,7,15,20,28,31,40)(5,12,17,23,29,36,42,6,11,18,24,30,35,41)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$21$:  $C_7:C_3$
$42$:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $S_3$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$

Degree 21: 21T11

Low degree siblings

21T11, 42T23

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $7$ $3$ $( 7,14,26)( 8,13,25)( 9,16,27)(10,15,28)(11,17,29)(12,18,30)(19,37,32) (20,38,31)(21,39,33)(22,40,34)(23,41,36)(24,42,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $7$ $3$ $( 7,26,14)( 8,25,13)( 9,27,16)(10,28,15)(11,29,17)(12,30,18)(19,32,37) (20,31,38)(21,33,39)(22,34,40)(23,36,41)(24,35,42)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23) (22,24)(25,26)(27,30)(28,29)(31,32)(33,36)(34,35)(37,38)(39,41)(40,42)$
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ $21$ $6$ $( 1, 2)( 3, 5)( 4, 6)( 7,13,26, 8,14,25)( 9,18,27,12,16,30)(10,17,28,11,15,29) (19,38,32,20,37,31)(21,41,33,23,39,36)(22,42,34,24,40,35)$
$ 6, 6, 6, 6, 6, 6, 2, 2, 2 $ $21$ $6$ $( 1, 2)( 3, 5)( 4, 6)( 7,25,14, 8,26,13)( 9,30,16,12,27,18)(10,29,15,11,28,17) (19,31,37,20,32,38)(21,36,39,23,33,41)(22,35,40,24,34,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,22,23) (20,21,24)(25,28,30)(26,27,29)(31,33,35)(32,34,36)(37,40,41)(38,39,42)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,16,29)( 8,15,30)( 9,17,26)(10,18,25)(11,14,27) (12,13,28)(19,40,36)(20,39,35)(21,42,31)(22,41,32)(23,37,34)(24,38,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $14$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7,27,17)( 8,28,18)( 9,29,14)(10,30,13)(11,26,16) (12,25,15)(19,34,41)(20,33,42)(21,35,38)(22,36,37)(23,32,40)(24,31,39)$
$ 14, 14, 14 $ $9$ $14$ $( 1, 7,13,20,25,31,37, 2, 8,14,19,26,32,38)( 3,11,15,24,28,35,40, 5,10,17,22, 29,34,42)( 4,12,16,23,27,36,39, 6, 9,18,21,30,33,41)$
$ 7, 7, 7, 7, 7, 7 $ $3$ $7$ $( 1, 8,13,19,25,32,37)( 2, 7,14,20,26,31,38)( 3,10,15,22,28,34,40) ( 4, 9,16,21,27,33,39)( 5,11,17,24,29,35,42)( 6,12,18,23,30,36,41)$
$ 21, 21 $ $6$ $21$ $( 1,10,18,19,28,36,37, 3,12,13,22,30,32,40, 6, 8,15,23,25,34,41) ( 2, 9,17,20,27,35,38, 4,11,14,21,29,31,39, 5, 7,16,24,26,33,42)$
$ 7, 7, 7, 7, 7, 7 $ $3$ $7$ $( 1,19,37,13,32, 8,25)( 2,20,38,14,31, 7,26)( 3,22,40,15,34,10,28) ( 4,21,39,16,33, 9,27)( 5,24,42,17,35,11,29)( 6,23,41,18,36,12,30)$
$ 14, 14, 14 $ $9$ $14$ $( 1,20,37,14,32, 7,25, 2,19,38,13,31, 8,26)( 3,24,40,17,34,11,28, 5,22,42,15, 35,10,29)( 4,23,39,18,33,12,27, 6,21,41,16,36, 9,30)$
$ 21, 21 $ $6$ $21$ $( 1,22,41,13,34,12,25, 3,23,37,15,36, 8,28, 6,19,40,18,32,10,30) ( 2,21,42,14,33,11,26, 4,24,38,16,35, 7,27, 5,20,39,17,31, 9,29)$

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:  126.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 7A1 7A-1 14A1 14A-1 21A1 21A-1
Size 1 3 2 7 7 14 14 21 21 3 3 9 9 6 6
2 P 1A 1A 3A 3B-1 3B1 3C-1 3C1 3B1 3B-1 7A1 7A-1 7A-1 7A1 21A1 21A-1
3 P 1A 2A 1A 1A 1A 1A 1A 2A 2A 7A-1 7A1 14A-1 14A1 7A1 7A-1
7 P 1A 2A 3A 3B1 3B-1 3C1 3C-1 6A1 6A-1 1A 1A 2A 2A 3A 3A
Type
126.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
126.8.1b R 1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1 1 1
126.8.1c1 C 1 1 1 ζ3−1 ζ3 ζ3 ζ3−1 ζ3 ζ3−1 1 1 1 1 1 1
126.8.1c2 C 1 1 1 ζ3 ζ3−1 ζ3−1 ζ3 ζ3−1 ζ3 1 1 1 1 1 1
126.8.1d1 C 1 −1 1 ζ3−1 ζ3 ζ3 ζ3−1 −ζ3 −ζ3−1 1 1 −1 −1 1 1
126.8.1d2 C 1 −1 1 ζ3 ζ3−1 ζ3−1 ζ3 −ζ3−1 −ζ3 1 1 −1 −1 1 1
126.8.2a R 2 0 −1 2 2 −1 −1 0 0 2 2 0 0 −1 −1
126.8.2b1 C 2 0 −1 2ζ3−1 2ζ3 −ζ3 −ζ3−1 0 0 2 2 0 0 −1 −1
126.8.2b2 C 2 0 −1 2ζ3 2ζ3−1 −ζ3−1 −ζ3 0 0 2 2 0 0 −1 −1
126.8.3a1 C 3 3 3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
126.8.3a2 C 3 3 3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
126.8.3b1 C 3 −3 3 0 0 0 0 0 0 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
126.8.3b2 C 3 −3 3 0 0 0 0 0 0 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
126.8.6a1 C 6 0 −3 0 0 0 0 0 0 −2ζ7−3−2−2ζ7−2ζ72 2ζ7−3+2ζ7+2ζ72 0 0 −ζ7−3−ζ7−ζ72 ζ7−3+1+ζ7+ζ72
126.8.6a2 C 6 0 −3 0 0 0 0 0 0 2ζ7−3+2ζ7+2ζ72 −2ζ7−3−2−2ζ7−2ζ72 0 0 ζ7−3+1+ζ7+ζ72 −ζ7−3−ζ7−ζ72

magma: CharacterTable(G);