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Properties

Label 42T41
Degree $42$
Order $252$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{42}:C_6$

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Show commands: Magma

magma: G := TransitiveGroup(42, 41);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{42}:C_6$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,42,19,15,29)(2,11,41,20,16,30)(3,10,38,23,17,28)(4,9,37,24,18,27)(5,7,39,21,14,26)(6,8,40,22,13,25)(31,32)(33,35)(34,36), (1,32,15,7,24,40)(2,31,16,8,23,39)(3,35,17,11,20,38)(4,36,18,12,19,37)(5,33,14,10,22,41)(6,34,13,9,21,42)(25,28)(26,27)
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
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$42$:  $F_7$
$84$:  $F_7 \times C_2$
$126$:  21T10

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 6: $D_{6}$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

Degree 21: 21T10

Low degree siblings

42T41

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  252.30
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);