# Properties

 Label 40T6 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times D_4$

Show commands: Magma

magma: G := TransitiveGroup(40, 6);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $6$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5\times D_4$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $2$ magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $40$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,29,17,8,36,23,10,39,28,14)(2,30,18,7,35,24,9,40,27,13)(3,31,20,5,33,22,11,38,25,15)(4,32,19,6,34,21,12,37,26,16), (1,34,27,20,10,4,35,25,17,12,2,33,28,19,9,3,36,26,18,11)(5,40,32,23,15,7,37,29,22,13,6,39,31,24,16,8,38,30,21,14) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $C_5$

Degree 8: $D_4$

Degree 10: $C_{10}$ x 3

Degree 20: 20T3, 20T12 x 2

## Low degree siblings

20T12 x 2

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $40=2^{3} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 40.10 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);