# Properties

 Label 40T48 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5:D_8$

Show commands: Magma

magma: G := TransitiveGroup(40, 48);

## Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $48$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5:D_8$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $10$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,14,7,19,11,2,13,8,20,12)(3,15,5,17,10)(4,16,6,18,9)(21,33,27,37,32,24,36,25,39,29)(22,34,28,38,31,23,35,26,40,30), (1,33,3,36,2,34,4,35)(5,32,8,30,6,31,7,29)(9,28,11,25,10,27,12,26)(13,24,15,21,14,23,16,22)(17,39,19,38,18,40,20,37) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

Degree 10: $D_5$

Degree 20: 20T11

## Low degree siblings

40T32

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

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: 80.15 magma: IdentifyGroup(G);
 Character table:  2 4 2 4 3 2 2 3 2 3 2 2 2 3 3 3 2 3 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . . 1a 2a 2b 4a 10a 20a 5a 10b 10c 20b 10d 10e 5b 10f 8a 2c 8b 2P 1a 1a 1a 2b 5b 10f 5b 5b 5b 10c 5a 5a 5a 5a 4a 1a 4a 3P 1a 2a 2b 4a 10e 20b 5b 10d 10f 20a 10a 10b 5a 10c 8b 2c 8a 5P 1a 2a 2b 4a 2a 4a 1a 2a 2b 4a 2a 2a 1a 2b 8b 2c 8a 7P 1a 2a 2b 4a 10d 20b 5b 10e 10f 20a 10b 10a 5a 10c 8a 2c 8b 11P 1a 2a 2b 4a 10a 20a 5a 10b 10c 20b 10d 10e 5b 10f 8b 2c 8a 13P 1a 2a 2b 4a 10e 20b 5b 10d 10f 20a 10a 10b 5a 10c 8b 2c 8a 17P 1a 2a 2b 4a 10d 20b 5b 10e 10f 20a 10b 10a 5a 10c 8a 2c 8b 19P 1a 2a 2b 4a 10b 20a 5a 10a 10c 20b 10e 10d 5b 10f 8b 2c 8a X.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 X.3 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 X.5 2 . 2 -2 . -2 2 . 2 -2 . . 2 2 . . . X.6 2 -2 2 2 A -A -A A -A -*A *A *A -*A -*A . . . X.7 2 -2 2 2 *A -*A -*A *A -*A -A A A -A -A . . . X.8 2 . -2 . . . 2 . -2 . . . 2 -2 E . -E X.9 2 . -2 . . . 2 . -2 . . . 2 -2 -E . E X.10 2 . 2 -2 B A -A -B -A *A C -C -*A -*A . . . X.11 2 . 2 -2 C *A -*A -C -*A A -B B -A -A . . . X.12 2 . 2 -2 -C *A -*A C -*A A B -B -A -A . . . X.13 2 . 2 -2 -B A -A B -A *A -C C -*A -*A . . . X.14 2 2 2 2 -*A -*A -*A -*A -*A -A -A -A -A -A . . . X.15 2 2 2 2 -A -A -A -A -A -*A -*A -*A -*A -*A . . . X.16 4 . -4 . . . D . -D . . . *D -*D . . . X.17 4 . -4 . . . *D . -*D . . . D -D . . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -E(5)+E(5)^4 C = -E(5)^2+E(5)^3 D = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 E = -E(8)+E(8)^3 = -Sqrt(2) = -r2 

magma: CharacterTable(G);