# Properties

 Label 40T41 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2^2\times D_{10}$

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magma: G := TransitiveGroup(40, 41);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $41$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_2^2\times D_{10}$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $8$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39), (1,32,19,7,35,23,12,40,26,16)(2,31,20,8,36,24,11,39,25,15)(3,30,17,5,34,22,10,37,27,14)(4,29,18,6,33,21,9,38,28,13), (1,11)(2,12)(3,9)(4,10)(5,6)(7,8)(13,37)(14,38)(15,40)(16,39)(17,33)(18,34)(19,36)(20,35)(21,30)(22,29)(23,31)(24,32)(25,26)(27,28), (1,6,12,13,19,21,26,29,35,38)(2,5,11,14,20,22,25,30,36,37)(3,8,10,15,17,24,27,31,34,39)(4,7,9,16,18,23,28,32,33,40) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$10$:  $D_{5}$
$16$:  $C_2^4$
$20$:  $D_{10}$ x 7
$40$:  20T8 x 7

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 5: $D_{5}$

Degree 8: $C_2^3$

Degree 10: $D_{10}$ x 7

Degree 20: 20T8 x 7

## Low degree siblings

40T41 x 7

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

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

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); Label: 80.51 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);