Properties

Label 40T19
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4:C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(40, 19);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $19$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:C_{10}$
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:  (3,4)(7,8)(11,12)(13,14)(19,20)(23,24)(25,26)(29,30)(33,34)(39,40), (1,30,18,7,35,23,10,39,28,13)(2,29,17,8,36,24,9,40,27,14)(3,31,19,5,34,21,11,37,25,15)(4,32,20,6,33,22,12,38,26,16), (1,34,28,19,10,3,35,25,18,11)(2,33,27,20,9,4,36,26,17,12)(5,40,31,24,15,8,37,29,21,14)(6,39,32,23,16,7,38,30,22,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$5$:  $C_5$
$8$:  $C_2^3$
$10$:  $C_{10}$ x 7
$16$:  $Q_8:C_2$
$20$:  20T3 x 7
$40$:  40T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 8: $Q_8:C_2$

Degree 10: $C_{10}$ x 3

Degree 20: 20T3

Low degree siblings

40T19 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, 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 $ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)(19,20)(23,24)(25,26)(29,30)(33,34)(39,40)$
$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$
$ 20, 20 $ $2$ $20$ $( 1, 5, 9,16,18,21,27,32,35,37, 2, 6,10,15,17,22,28,31,36,38)( 3, 7,12,14,19, 23,26,29,34,39, 4, 8,11,13,20,24,25,30,33,40)$
$ 20, 20 $ $1$ $20$ $( 1, 5, 9,16,18,21,27,32,35,37, 2, 6,10,15,17,22,28,31,36,38)( 3, 8,12,13,19, 24,26,30,34,40, 4, 7,11,14,20,23,25,29,33,39)$
$ 20, 20 $ $1$ $20$ $( 1, 6, 9,15,18,22,27,31,35,38, 2, 5,10,16,17,21,28,32,36,37)( 3, 7,12,14,19, 23,26,29,34,39, 4, 8,11,13,20,24,25,30,33,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1, 7,10,13,18,23,28,30,35,39)( 2, 8, 9,14,17,24,27,29,36,40)( 3, 5,11,15,19, 21,25,31,34,37)( 4, 6,12,16,20,22,26,32,33,38)$
$ 20, 20 $ $2$ $20$ $( 1, 7, 9,14,18,23,27,29,35,39, 2, 8,10,13,17,24,28,30,36,40)( 3, 6,12,15,19, 22,26,31,34,38, 4, 5,11,16,20,21,25,32,33,37)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1, 9,18,27,35, 2,10,17,28,36)( 3,11,19,25,34)( 4,12,20,26,33) ( 5,16,21,32,37, 6,15,22,31,38)( 7,13,23,30,39)( 8,14,24,29,40)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1, 9,18,27,35, 2,10,17,28,36)( 3,12,19,26,34, 4,11,20,25,33)( 5,16,21,32,37, 6,15,22,31,38)( 7,14,23,29,39, 8,13,24,30,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,10,18,28,35)( 2, 9,17,27,36)( 3,11,19,25,34)( 4,12,20,26,33) ( 5,15,21,31,37)( 6,16,22,32,38)( 7,13,23,30,39)( 8,14,24,29,40)$
$ 20, 20 $ $2$ $20$ $( 1,11,17,26,35, 3, 9,20,28,34, 2,12,18,25,36, 4,10,19,27,33)( 5,14,22,30,37, 8,16,23,31,40, 6,13,21,29,38, 7,15,24,32,39)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,11,18,25,35, 3,10,19,28,34)( 2,12,17,26,36, 4, 9,20,27,33)( 5,14,21,29,37, 8,15,24,31,40)( 6,13,22,30,38, 7,16,23,32,39)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,13,28,39,10,23,35, 7,18,30)( 2,14,27,40, 9,24,36, 8,17,29)( 3,15,25,37,11, 21,34, 5,19,31)( 4,16,26,38,12,22,33, 6,20,32)$
$ 20, 20 $ $2$ $20$ $( 1,13,27,40,10,23,36, 8,18,30, 2,14,28,39, 9,24,35, 7,17,29)( 3,16,26,37,11, 22,33, 5,19,32, 4,15,25,38,12,21,34, 6,20,31)$
$ 20, 20 $ $2$ $20$ $( 1,15,27,38,10,21,36, 6,18,31, 2,16,28,37, 9,22,35, 5,17,32)( 3,13,26,40,11, 23,33, 8,19,30, 4,14,25,39,12,24,34, 7,20,29)$
$ 20, 20 $ $1$ $20$ $( 1,15,27,38,10,21,36, 6,18,31, 2,16,28,37, 9,22,35, 5,17,32)( 3,14,26,39,11, 24,33, 7,19,29, 4,13,25,40,12,23,34, 8,20,30)$
$ 20, 20 $ $1$ $20$ $( 1,16,27,37,10,22,36, 5,18,32, 2,15,28,38, 9,21,35, 6,17,31)( 3,13,26,40,11, 23,33, 8,19,30, 4,14,25,39,12,24,34, 7,20,29)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,17,35, 9,28, 2,18,36,10,27)( 3,19,34,11,25)( 4,20,33,12,26) ( 5,22,37,16,31, 6,21,38,15,32)( 7,23,39,13,30)( 8,24,40,14,29)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,17,35, 9,28, 2,18,36,10,27)( 3,20,34,12,25, 4,19,33,11,26)( 5,22,37,16,31, 6,21,38,15,32)( 7,24,39,14,30, 8,23,40,13,29)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,18,35,10,28)( 2,17,36, 9,27)( 3,19,34,11,25)( 4,20,33,12,26) ( 5,21,37,15,31)( 6,22,38,16,32)( 7,23,39,13,30)( 8,24,40,14,29)$
$ 20, 20 $ $2$ $20$ $( 1,19,36,12,28, 3,17,33,10,25, 2,20,35,11,27, 4,18,34, 9,26)( 5,24,38,13,31, 8,22,39,15,29, 6,23,37,14,32, 7,21,40,16,30)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,35,11,28, 3,18,34,10,25)( 2,20,36,12,27, 4,17,33, 9,26)( 5,24,37,14,31, 8,21,40,15,29)( 6,23,38,13,32, 7,22,39,16,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 2,22)( 3,23, 4,24)( 5,27, 6,28)( 7,26, 8,25)( 9,32,10,31)(11,30,12,29) (13,33,14,34)(15,36,16,35)(17,38,18,37)(19,39,20,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,21, 2,22)( 3,24, 4,23)( 5,27, 6,28)( 7,25, 8,26)( 9,32,10,31)(11,29,12,30) (13,34,14,33)(15,36,16,35)(17,38,18,37)(19,40,20,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,22, 2,21)( 3,23, 4,24)( 5,28, 6,27)( 7,26, 8,25)( 9,31,10,32)(11,30,12,29) (13,33,14,34)(15,35,16,36)(17,37,18,38)(19,39,20,40)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,25)( 6,26)( 7,28)( 8,27)( 9,29)(10,30)(11,31) (12,32)(13,35)(14,36)(15,34)(16,33)(17,40)(18,39)(19,37)(20,38)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,23, 2,24)( 3,22, 4,21)( 5,25, 6,26)( 7,27, 8,28)( 9,29,10,30)(11,32,12,31) (13,36,14,35)(15,34,16,33)(17,40,18,39)(19,38,20,37)$
$ 20, 20 $ $2$ $20$ $( 1,25, 9,33,18, 3,27,12,35,19, 2,26,10,34,17, 4,28,11,36,20)( 5,29,16,39,21, 8,32,13,37,24, 6,30,15,40,22, 7,31,14,38,23)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,25,10,34,18, 3,28,11,35,19)( 2,26, 9,33,17, 4,27,12,36,20)( 5,29,15,40,21, 8,31,14,37,24)( 6,30,16,39,22, 7,32,13,38,23)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,27,10,36,18, 2,28, 9,35,17)( 3,25,11,34,19)( 4,26,12,33,20) ( 5,32,15,38,21, 6,31,16,37,22)( 7,30,13,39,23)( 8,29,14,40,24)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,27,10,36,18, 2,28, 9,35,17)( 3,26,11,33,19, 4,25,12,34,20)( 5,32,15,38,21, 6,31,16,37,22)( 7,29,13,40,23, 8,30,14,39,24)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,28,10,35,18)( 2,27, 9,36,17)( 3,25,11,34,19)( 4,26,12,33,20) ( 5,31,15,37,21)( 6,32,16,38,22)( 7,30,13,39,23)( 8,29,14,40,24)$
$ 20, 20 $ $2$ $20$ $( 1,29,17, 7,35,24, 9,39,28,14, 2,30,18, 8,36,23,10,40,27,13)( 3,31,20, 6,34, 21,12,38,25,15, 4,32,19, 5,33,22,11,37,26,16)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,29,18, 8,35,24,10,40,28,14)( 2,30,17, 7,36,23, 9,39,27,13)( 3,32,19, 6,34, 22,11,38,25,16)( 4,31,20, 5,33,21,12,37,26,15)$
$ 20, 20 $ $1$ $20$ $( 1,31,17, 6,35,21, 9,38,28,15, 2,32,18, 5,36,22,10,37,27,16)( 3,29,20, 7,34, 24,12,39,25,14, 4,30,19, 8,33,23,11,40,26,13)$
$ 20, 20 $ $2$ $20$ $( 1,31,17, 6,35,21, 9,38,28,15, 2,32,18, 5,36,22,10,37,27,16)( 3,30,20, 8,34, 23,12,40,25,13, 4,29,19, 7,33,24,11,39,26,14)$
$ 20, 20 $ $1$ $20$ $( 1,32,17, 5,35,22, 9,37,28,16, 2,31,18, 6,36,21,10,38,27,15)( 3,30,20, 8,34, 23,12,40,25,13, 4,29,19, 7,33,24,11,39,26,14)$
$ 20, 20 $ $2$ $20$ $( 1,33,27,19,10, 4,36,25,18,12, 2,34,28,20, 9, 3,35,26,17,11)( 5,39,32,24,15, 7,38,29,21,13, 6,40,31,23,16, 8,37,30,22,14)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,33,28,20,10, 4,35,26,18,12)( 2,34,27,19, 9, 3,36,25,17,11)( 5,39,31,23,15, 7,37,30,21,13)( 6,40,32,24,16, 8,38,29,22,14)$
$ 10, 10, 5, 5, 5, 5 $ $2$ $10$ $( 1,35,28,18,10)( 2,36,27,17, 9)( 3,33,25,20,11, 4,34,26,19,12) ( 5,37,31,21,15)( 6,38,32,22,16)( 7,40,30,24,13, 8,39,29,23,14)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $1$ $5$ $( 1,35,28,18,10)( 2,36,27,17, 9)( 3,34,25,19,11)( 4,33,26,20,12) ( 5,37,31,21,15)( 6,38,32,22,16)( 7,39,30,23,13)( 8,40,29,24,14)$
$ 10, 10, 10, 10 $ $1$ $10$ $( 1,36,28,17,10, 2,35,27,18, 9)( 3,33,25,20,11, 4,34,26,19,12)( 5,38,31,22,15, 6,37,32,21,16)( 7,40,30,24,13, 8,39,29,23,14)$
$ 20, 20 $ $2$ $20$ $( 1,37,36,32,28,21,17,16,10, 5, 2,38,35,31,27,22,18,15, 9, 6)( 3,39,33,29,25, 23,20,14,11, 7, 4,40,34,30,26,24,19,13,12, 8)$
$ 20, 20 $ $1$ $20$ $( 1,37,36,32,28,21,17,16,10, 5, 2,38,35,31,27,22,18,15, 9, 6)( 3,40,33,30,25, 24,20,13,11, 8, 4,39,34,29,26,23,19,14,12, 7)$
$ 20, 20 $ $1$ $20$ $( 1,38,36,31,28,22,17,15,10, 6, 2,37,35,32,27,21,18,16, 9, 5)( 3,39,33,29,25, 23,20,14,11, 7, 4,40,34,30,26,24,19,13,12, 8)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,39,35,30,28,23,18,13,10, 7)( 2,40,36,29,27,24,17,14, 9, 8)( 3,37,34,31,25, 21,19,15,11, 5)( 4,38,33,32,26,22,20,16,12, 6)$
$ 20, 20 $ $2$ $20$ $( 1,39,36,29,28,23,17,14,10, 7, 2,40,35,30,27,24,18,13, 9, 8)( 3,38,33,31,25, 22,20,15,11, 6, 4,37,34,32,26,21,19,16,12, 5)$

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:  80.48
magma: IdentifyGroup(G);
 
Character table:    50 x 50 character table

magma: CharacterTable(G);