# Properties

 Label 40T34 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $Q_8\times D_5$

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magma: G := TransitiveGroup(40, 34);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $34$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $Q_8\times D_5$ 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: (1,24,2,23)(3,21,4,22)(5,17,6,18)(7,20,8,19)(9,13,10,14)(11,15,12,16)(25,39,26,40)(27,37,28,38)(29,33,30,34)(31,36,32,35), (1,18,2,17)(3,19,4,20)(5,16,6,15)(7,13,8,14)(9,12,10,11)(21,40,22,39)(23,37,24,38)(25,34,26,33)(27,35,28,36)(29,32,30,31), (1,15,26,39,12,23,35,7,19,31,2,16,25,40,11,24,36,8,20,32)(3,14,27,38,9,22,34,6,17,29,4,13,28,37,10,21,33,5,18,30) 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
$8$:  $C_2^3$, $Q_8$ x 2
$10$:  $D_{5}$
$16$:  $Q_8\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8

## Low degree siblings

40T34

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

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: 80.41 magma: IdentifyGroup(G);
 Character table:  2 4 4 4 4 3 3 3 2 3 2 2 3 3 2 2 2 3 3 3 3 5 1 . 1 . 1 . . 1 . 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a 5a 20d 20e 20f 5b 10b 4e 4f 2P 1a 1a 1a 1a 2b 2b 2b 10a 2b 10a 10b 5b 5b 10b 10b 10a 5a 5a 2b 2b 3P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b 5b 20a 20b 20c 5a 10a 4e 4f 5P 1a 2a 2b 2c 4a 4b 4c 4e 4d 4f 4a 2b 1a 4e 4f 4a 1a 2b 4e 4f 7P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b 5b 20a 20b 20c 5a 10a 4e 4f 11P 1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a 5a 20d 20e 20f 5b 10b 4e 4f 13P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b 5b 20a 20b 20c 5a 10a 4e 4f 17P 1a 2a 2b 2c 4a 4b 4c 20d 4d 20e 20f 10b 5b 20a 20b 20c 5a 10a 4e 4f 19P 1a 2a 2b 2c 4a 4b 4c 20a 4d 20b 20c 10a 5a 20d 20e 20f 5b 10b 4e 4f X.1 1 1 1 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 1 1 1 -1 X.3 1 -1 1 -1 -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 1 1 1 1 X.5 1 -1 1 -1 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 1 1 -1 1 X.7 1 1 1 1 -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 1 1 -1 -1 X.9 2 -2 -2 2 . . . . . . . -2 2 . . . 2 -2 . . X.10 2 2 -2 -2 . . . . . . . -2 2 . . . 2 -2 . . X.11 2 . 2 . -2 . . A . -A *A -*A -*A *A -*A A -A -A -2 2 X.12 2 . 2 . -2 . . *A . -*A A -A -A A -A *A -*A -*A -2 2 X.13 2 . 2 . -2 . . -*A . *A A -A -A -A A *A -*A -*A 2 -2 X.14 2 . 2 . -2 . . -A . A *A -*A -*A -*A *A A -A -A 2 -2 X.15 2 . 2 . 2 . . A . A -*A -*A -*A *A *A -A -A -A -2 -2 X.16 2 . 2 . 2 . . *A . *A -A -A -A A A -*A -*A -*A -2 -2 X.17 2 . 2 . 2 . . -*A . -*A -A -A -A -A -A -*A -*A -*A 2 2 X.18 2 . 2 . 2 . . -A . -A -*A -*A -*A -*A -*A -A -A -A 2 2 X.19 4 . -4 . . . . . . . . B -B . . . -*B *B . . X.20 4 . -4 . . . . . . . . *B -*B . . . -B B . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -2*E(5)^2-2*E(5)^3 = 1+Sqrt(5) = 1+r5 

magma: CharacterTable(G);