LMFDB - Galois group: 42T40

Show commands: Magma

magma: G := TransitiveGroup(42, 40);

Group action invariants

Degree $n$:	$42$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$40$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_6\times F_7$
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:	(1,36,39,16,30,22)(2,35,40,15,29,21)(3,32,41,18,26,24)(4,31,42,17,25,23)(5,33,37,14,28,20)(6,34,38,13,27,19)(7,9,12,8,10,11), (7,22,14,40,25,32)(8,21,13,39,26,31)(9,23,15,41,28,34)(10,24,16,42,27,33)(11,19,17,37,30,35)(12,20,18,38,29,36)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$ x 4
$4$: $C_2^2$
$6$: $C_6$ x 12
$9$: $C_3^2$
$12$: $C_6\times C_2$ x 4
$18$: $C_6 \times C_3$ x 3
$36$: 36T4
$42$: $F_7$
$84$: $F_7 \times C_2$
$126$: 21T9

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$ x 4
$4$:	$C_2^2$
$6$:	$C_6$ x 12
$9$:	$C_3^2$
$12$:	$C_6\times C_2$ x 4
$18$:	$C_6 \times C_3$ x 3
$36$:	36T4
$42$:	$F_7$
$84$:	$F_7 \times C_2$
$126$:	21T9

Subfields

Degree 2: $C_2$
Degree 3: $C_3$
Degree 6: $C_6$
Degree 7: $F_7$
Degree 14: $F_7 \times C_2$
Degree 21: 21T9

Low degree siblings

42T40 x 5
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:	252.28	magma: IdentifyGroup(G);
Character table:	42 x 42 character table

magma: CharacterTable(G);

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	42T40
Degree	$42$
Order	$252$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_6\times F_7$