LMFDB - Galois group: 40T16

Show commands: Magma

magma: G := TransitiveGroup(40, 16);

Group action invariants

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

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$5$: $C_5$
$8$: $D_{4}$ x 2, $C_4\times C_2$
$10$: $C_{10}$ x 3
$16$: $C_2^2:C_4$
$20$: 20T1 x 2, 20T3
$40$: 20T12 x 2, 40T2

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$5$:	$C_5$
$8$:	$D_{4}$ x 2, $C_4\times C_2$
$10$:	$C_{10}$ x 3
$16$:	$C_2^2:C_4$
$20$:	20T1 x 2, 20T3
$40$:	20T12 x 2, 40T2

Subfields

Degree 2: $C_2$
Degree 4: $C_4$, $D_{4}$ x 2
Degree 5: $C_5$
Degree 8: $C_2^2:C_4$
Degree 10: $C_{10}$
Degree 20: 20T1, 20T12 x 2

Low degree siblings

40T16
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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$2$	$( 5, 6)( 7, 8)(13,14)(15,16)(21,22)(23,24)(29,30)(31,32)(37,38)(39,40)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$2$	$2$	$( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,39)(38,40)$
	$ 20, 20 $	$2$	$20$	$( 1, 5,11,13,18,22,25,29,35,37, 3, 7, 9,15,20,24,28,31,34,40)( 2, 6,12,14,17, 21,26,30,36,38, 4, 8,10,16,19,23,27,32,33,39)$
	$ 20, 20 $	$2$	$20$	$( 1, 5,12,14,18,22,26,30,35,37, 4, 8, 9,15,19,23,28,31,33,39)( 2, 6,11,13,17, 21,25,29,36,38, 3, 7,10,16,20,24,27,32,34,40)$
	$ 20, 20 $	$2$	$20$	$( 1, 7,11,15,18,24,25,31,35,40, 3, 5, 9,13,20,22,28,29,34,37)( 2, 8,12,16,17, 23,26,32,36,39, 4, 6,10,14,19,21,27,30,33,38)$
	$ 20, 20 $	$2$	$20$	$( 1, 7,12,16,18,24,26,32,35,40, 4, 6, 9,13,19,21,28,29,33,38)( 2, 8,11,15,17, 23,25,31,36,39, 3, 5,10,14,20,22,27,30,34,37)$
	$ 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1, 9,18,28,35)( 2,10,17,27,36)( 3,11,20,25,34)( 4,12,19,26,33) ( 5,15,22,31,37)( 6,16,21,32,38)( 7,13,24,29,40)( 8,14,23,30,39)$
	$ 10, 10, 5, 5, 5, 5 $	$2$	$10$	$( 1, 9,18,28,35)( 2,10,17,27,36)( 3,11,20,25,34)( 4,12,19,26,33) ( 5,16,22,32,37, 6,15,21,31,38)( 7,14,24,30,40, 8,13,23,29,39)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,10,18,27,35, 2, 9,17,28,36)( 3,12,20,26,34, 4,11,19,25,33)( 5,16,22,32,37, 6,15,21,31,38)( 7,14,24,30,40, 8,13,23,29,39)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,11,18,25,35, 3, 9,20,28,34)( 2,12,17,26,36, 4,10,19,27,33)( 5,13,22,29,37, 7,15,24,31,40)( 6,14,21,30,38, 8,16,23,32,39)$
	$ 10, 10, 10, 10 $	$2$	$10$	$( 1,11,18,25,35, 3, 9,20,28,34)( 2,12,17,26,36, 4,10,19,27,33)( 5,14,22,30,37, 8,15,23,31,39)( 6,13,21,29,38, 7,16,24,32,40)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,12,18,26,35, 4, 9,19,28,33)( 2,11,17,25,36, 3,10,20,27,34)( 5,14,22,30,37, 8,15,23,31,39)( 6,13,21,29,38, 7,16,24,32,40)$
	$ 20, 20 $	$2$	$20$	$( 1,13,26,38, 9,24,33, 6,18,29, 4,16,28,40,12,21,35, 7,19,32)( 2,14,25,37,10, 23,34, 5,17,30, 3,15,27,39,11,22,36, 8,20,31)$
	$ 20, 20 $	$2$	$20$	$( 1,13,25,37, 9,24,34, 5,18,29, 3,15,28,40,11,22,35, 7,20,31)( 2,14,26,38,10, 23,33, 6,17,30, 4,16,27,39,12,21,36, 8,19,32)$
	$ 20, 20 $	$2$	$20$	$( 1,15,26,39, 9,22,33, 8,18,31, 4,14,28,37,12,23,35, 5,19,30)( 2,16,25,40,10, 21,34, 7,17,32, 3,13,27,38,11,24,36, 6,20,29)$
	$ 20, 20 $	$2$	$20$	$( 1,15,25,40, 9,22,34, 7,18,31, 3,13,28,37,11,24,35, 5,20,29)( 2,16,26,39,10, 21,33, 8,17,32, 4,14,27,38,12,23,36, 6,19,30)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,17,35,10,28, 2,18,36, 9,27)( 3,19,34,12,25, 4,20,33,11,26)( 5,21,37,16,31, 6,22,38,15,32)( 7,23,40,14,29, 8,24,39,13,30)$
	$ 10, 10, 5, 5, 5, 5 $	$2$	$10$	$( 1,17,35,10,28, 2,18,36, 9,27)( 3,19,34,12,25, 4,20,33,11,26)( 5,22,37,15,31) ( 6,21,38,16,32)( 7,24,40,13,29)( 8,23,39,14,30)$
	$ 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,18,35, 9,28)( 2,17,36,10,27)( 3,20,34,11,25)( 4,19,33,12,26) ( 5,22,37,15,31)( 6,21,38,16,32)( 7,24,40,13,29)( 8,23,39,14,30)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,19,35,12,28, 4,18,33, 9,26)( 2,20,36,11,27, 3,17,34,10,25)( 5,23,37,14,31, 8,22,39,15,30)( 6,24,38,13,32, 7,21,40,16,29)$
	$ 10, 10, 10, 10 $	$2$	$10$	$( 1,19,35,12,28, 4,18,33, 9,26)( 2,20,36,11,27, 3,17,34,10,25)( 5,24,37,13,31, 7,22,40,15,29)( 6,23,38,14,32, 8,21,39,16,30)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,20,35,11,28, 3,18,34, 9,25)( 2,19,36,12,27, 4,17,33,10,26)( 5,24,37,13,31, 7,22,40,15,29)( 6,23,38,14,32, 8,21,39,16,30)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,21, 4,24)( 2,22, 3,23)( 5,25, 8,27)( 6,26, 7,28)( 9,32,12,29)(10,31,11,30) (13,35,16,33)(14,36,15,34)(17,37,20,39)(18,38,19,40)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,21, 3,23)( 2,22, 4,24)( 5,26, 7,27)( 6,25, 8,28)( 9,32,11,30)(10,31,12,29) (13,36,15,33)(14,35,16,34)(17,37,19,40)(18,38,20,39)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,23, 3,21)( 2,24, 4,22)( 5,27, 7,26)( 6,28, 8,25)( 9,30,11,32)(10,29,12,31) (13,33,15,36)(14,34,16,35)(17,40,19,37)(18,39,20,38)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$2$	$4$	$( 1,23, 4,22)( 2,24, 3,21)( 5,28, 8,26)( 6,27, 7,25)( 9,30,12,31)(10,29,11,32) (13,34,16,36)(14,33,15,35)(17,40,20,38)(18,39,19,37)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,25, 9,34,18, 3,28,11,35,20)( 2,26,10,33,17, 4,27,12,36,19)( 5,29,15,40,22, 7,31,13,37,24)( 6,30,16,39,21, 8,32,14,38,23)$
	$ 10, 10, 10, 10 $	$2$	$10$	$( 1,25, 9,34,18, 3,28,11,35,20)( 2,26,10,33,17, 4,27,12,36,19)( 5,30,15,39,22, 8,31,14,37,23)( 6,29,16,40,21, 7,32,13,38,24)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,26, 9,33,18, 4,28,12,35,19)( 2,25,10,34,17, 3,27,11,36,20)( 5,30,15,39,22, 8,31,14,37,23)( 6,29,16,40,21, 7,32,13,38,24)$
	$ 10, 10, 5, 5, 5, 5 $	$2$	$10$	$( 1,27, 9,36,18, 2,28,10,35,17)( 3,26,11,33,20, 4,25,12,34,19)( 5,31,15,37,22) ( 6,32,16,38,21)( 7,29,13,40,24)( 8,30,14,39,23)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,27, 9,36,18, 2,28,10,35,17)( 3,26,11,33,20, 4,25,12,34,19)( 5,32,15,38,22, 6,31,16,37,21)( 7,30,13,39,24, 8,29,14,40,23)$
	$ 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,28, 9,35,18)( 2,27,10,36,17)( 3,25,11,34,20)( 4,26,12,33,19) ( 5,31,15,37,22)( 6,32,16,38,21)( 7,29,13,40,24)( 8,30,14,39,23)$
	$ 20, 20 $	$2$	$20$	$( 1,29,20, 5,35,24,11,37,28,13, 3,31,18, 7,34,22, 9,40,25,15)( 2,30,19, 6,36, 23,12,38,27,14, 4,32,17, 8,33,21,10,39,26,16)$
	$ 20, 20 $	$2$	$20$	$( 1,29,19, 6,35,24,12,38,28,13, 4,32,18, 7,33,21, 9,40,26,16)( 2,30,20, 5,36, 23,11,37,27,14, 3,31,17, 8,34,22,10,39,25,15)$
	$ 20, 20 $	$2$	$20$	$( 1,31,19, 8,35,22,12,39,28,15, 4,30,18, 5,33,23, 9,37,26,14)( 2,32,20, 7,36, 21,11,40,27,16, 3,29,17, 6,34,24,10,38,25,13)$
	$ 20, 20 $	$2$	$20$	$( 1,31,20, 7,35,22,11,40,28,15, 3,29,18, 5,34,24, 9,37,25,13)( 2,32,19, 8,36, 21,12,39,27,16, 4,30,17, 6,33,23,10,38,26,14)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,33,28,19, 9, 4,35,26,18,12)( 2,34,27,20,10, 3,36,25,17,11)( 5,39,31,23,15, 8,37,30,22,14)( 6,40,32,24,16, 7,38,29,21,13)$
	$ 10, 10, 10, 10 $	$2$	$10$	$( 1,33,28,19, 9, 4,35,26,18,12)( 2,34,27,20,10, 3,36,25,17,11)( 5,40,31,24,15, 7,37,29,22,13)( 6,39,32,23,16, 8,38,30,21,14)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,34,28,20, 9, 3,35,25,18,11)( 2,33,27,19,10, 4,36,26,17,12)( 5,40,31,24,15, 7,37,29,22,13)( 6,39,32,23,16, 8,38,30,21,14)$
	$ 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,35,28,18, 9)( 2,36,27,17,10)( 3,34,25,20,11)( 4,33,26,19,12) ( 5,37,31,22,15)( 6,38,32,21,16)( 7,40,29,24,13)( 8,39,30,23,14)$
	$ 10, 10, 5, 5, 5, 5 $	$2$	$10$	$( 1,35,28,18, 9)( 2,36,27,17,10)( 3,34,25,20,11)( 4,33,26,19,12) ( 5,38,31,21,15, 6,37,32,22,16)( 7,39,29,23,13, 8,40,30,24,14)$
	$ 10, 10, 10, 10 $	$1$	$10$	$( 1,36,28,17, 9, 2,35,27,18,10)( 3,33,25,19,11, 4,34,26,20,12)( 5,38,31,21,15, 6,37,32,22,16)( 7,39,29,23,13, 8,40,30,24,14)$
	$ 20, 20 $	$2$	$20$	$( 1,37,34,29,28,22,20,13, 9, 5, 3,40,35,31,25,24,18,15,11, 7)( 2,38,33,30,27, 21,19,14,10, 6, 4,39,36,32,26,23,17,16,12, 8)$
	$ 20, 20 $	$2$	$20$	$( 1,37,33,30,28,22,19,14, 9, 5, 4,39,35,31,26,23,18,15,12, 8)( 2,38,34,29,27, 21,20,13,10, 6, 3,40,36,32,25,24,17,16,11, 7)$
	$ 20, 20 $	$2$	$20$	$( 1,39,33,31,28,23,19,15, 9, 8, 4,37,35,30,26,22,18,14,12, 5)( 2,40,34,32,27, 24,20,16,10, 7, 3,38,36,29,25,21,17,13,11, 6)$
	$ 20, 20 $	$2$	$20$	$( 1,39,34,32,28,23,20,16, 9, 8, 3,38,35,30,25,21,18,14,11, 6)( 2,40,33,31,27, 24,19,15,10, 7, 4,37,36,29,26,22,17,13,12, 5)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$80=2^{4} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	$2$
Label:	80.21	magma: IdentifyGroup(G);
Character table:	50 x 50 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	40T16
Degree	$40$
Order	$80$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_2^2:C_{20}$