# Properties

 Label 40T10 Degree $40$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_2\times D_{10}$

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magma: G := TransitiveGroup(40, 10);

## Group action invariants

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

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: $D_5$, $D_{10}$ x 6

Degree 20: 20T4 x 3, 20T8 x 4

## Low degree siblings

20T8 x 4

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

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: 40.13 magma: IdentifyGroup(G);
 Character table:  2 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 3 5 1 1 . . . . 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 2d 2e 10a 10b 5a 10c 10d 10e 5b 10f 2f 2g 2P 1a 1a 1a 1a 1a 1a 5a 5a 5b 5b 5b 5b 5a 5a 1a 1a 3P 1a 2a 2b 2c 2d 2e 10d 10e 5b 10f 10a 10b 5a 10c 2f 2g 5P 1a 2a 2b 2c 2d 2e 2g 2f 1a 2a 2g 2f 1a 2a 2f 2g 7P 1a 2a 2b 2c 2d 2e 10d 10e 5b 10f 10a 10b 5a 10c 2f 2g X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 -1 1 X.3 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 X.4 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 X.6 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.9 2 -2 . . . . A -A -*A *A *A -*A -A A 2 -2 X.10 2 -2 . . . . *A -*A -A A A -A -*A *A 2 -2 X.11 2 -2 . . . . -*A *A -A A -A A -*A *A -2 2 X.12 2 -2 . . . . -A A -*A *A -*A *A -A A -2 2 X.13 2 2 . . . . A A -*A -*A *A *A -A -A -2 -2 X.14 2 2 . . . . *A *A -A -A A A -*A -*A -2 -2 X.15 2 2 . . . . -*A -*A -A -A -A -A -*A -*A 2 2 X.16 2 2 . . . . -A -A -*A -*A -*A -*A -A -A 2 2 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 

magma: CharacterTable(G);