Properties

 Label 40T45 Degree $40$ Order $80$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{10}:C_4$

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magma: G := TransitiveGroup(40, 45);

Group action invariants

 Degree $n$: $40$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $45$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_{10}:C_4$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,39,27,31)(2,40,28,32)(3,37,25,29)(4,38,26,30)(5,12,21,18)(6,11,22,17)(7,9,24,20)(8,10,23,19)(13,34,15,36)(14,33,16,35), (1,22,4,24)(2,21,3,23)(5,11,39,34)(6,12,40,33)(7,10,37,36)(8,9,38,35)(13,27,31,20)(14,28,32,19)(15,25,29,18)(16,26,30,17) magma: Generators(G);

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 5: $F_5$

Degree 8: $C_2^2:C_4$

Degree 10: $F_5$

Degree 20: 20T9, 20T22 x 2

Low degree siblings

20T19 x 2, 20T22 x 2, 40T26, 40T55 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $10$ $2$ $( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,25)(20,26)(21,22)(23,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,40)(38,39)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 3)( 2, 4)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,33)(12,34)(13,32) (14,31)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,24)(22,23)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $5$ $2$ $( 1, 4)( 2, 3)( 5,39)( 6,40)( 7,37)( 8,38)( 9,35)(10,36)(11,34)(12,33)(13,31) (14,32)(15,29)(16,30)(17,26)(18,25)(19,28)(20,27)(21,23)(22,24)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 5,18,14)( 2, 6,17,13)( 3, 7,20,16)( 4, 8,19,15)( 9,29,11,32)(10,30,12,31) (21,26,37,33)(22,25,38,34)(23,27,40,36)(24,28,39,35)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 5,33,30)( 2, 6,34,29)( 3, 7,35,32)( 4, 8,36,31)( 9,22,26,16)(10,21,25,15) (11,24,27,13)(12,23,28,14)(17,40,19,38)(18,39,20,37)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 7,17,15)( 2, 8,18,16)( 3, 5,19,13)( 4, 6,20,14)( 9,31,12,29)(10,32,11,30) (21,27,38,35)(22,28,37,36)(23,26,39,34)(24,25,40,33)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 4$ $10$ $4$ $( 1, 7,34,31)( 2, 8,33,32)( 3, 5,36,29)( 4, 6,35,30)( 9,23,25,13)(10,24,26,14) (11,21,28,16)(12,22,27,15)(17,38,20,39)(18,37,19,40)$ $10, 10, 10, 10$ $4$ $10$ $( 1, 9,19,28,35, 4,11,18,25,34)( 2,10,20,27,36, 3,12,17,26,33)( 5,13,24,30,38, 7,15,21,32,40)( 6,14,23,29,37, 8,16,22,31,39)$ $10, 10, 10, 10$ $4$ $10$ $( 1,10,19,27,35, 3,11,17,25,33)( 2, 9,20,28,36, 4,12,18,26,34)( 5,14,24,29,38, 8,15,22,32,39)( 6,13,23,30,37, 7,16,21,31,40)$ $5, 5, 5, 5, 5, 5, 5, 5$ $4$ $5$ $( 1,11,19,25,35)( 2,12,20,26,36)( 3,10,17,27,33)( 4, 9,18,28,34) ( 5,15,24,32,38)( 6,16,23,31,37)( 7,13,21,30,40)( 8,14,22,29,39)$ $10, 10, 10, 10$ $4$ $10$ $( 1,12,19,26,35, 2,11,20,25,36)( 3, 9,17,28,33, 4,10,18,27,34)( 5,16,24,31,38, 6,15,23,32,37)( 7,14,21,29,40, 8,13,22,30,39)$

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: 80.34 magma: IdentifyGroup(G);
 Character table:  2 4 3 4 3 4 4 3 3 3 3 2 2 2 2 5 1 . 1 1 . . . . . . 1 1 1 1 1a 2a 2b 2c 2d 2e 4a 4b 4c 4d 10a 10b 5a 10c 2P 1a 1a 1a 1a 1a 1a 2e 2d 2d 2e 5a 5a 5a 5a 3P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10b 10a 5a 10c 5P 1a 2a 2b 2c 2d 2e 4a 4b 4c 4d 2c 2c 1a 2b 7P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10b 10a 5a 10c X.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 X.3 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 X.5 1 -1 1 1 -1 -1 A -A A -A 1 1 1 1 X.6 1 -1 1 1 -1 -1 -A A -A A 1 1 1 1 X.7 1 1 1 -1 -1 -1 A A -A -A -1 -1 1 1 X.8 1 1 1 -1 -1 -1 -A -A A A -1 -1 1 1 X.9 2 . -2 . -2 2 . . . . . . 2 -2 X.10 2 . -2 . 2 -2 . . . . . . 2 -2 X.11 4 . 4 -4 . . . . . . 1 1 -1 -1 X.12 4 . 4 4 . . . . . . -1 -1 -1 -1 X.13 4 . -4 . . . . . . . B -B -1 1 X.14 4 . -4 . . . . . . . -B B -1 1 A = -E(4) = -Sqrt(-1) = -i B = -E(5)+E(5)^2+E(5)^3-E(5)^4 = -Sqrt(5) = -r5 

magma: CharacterTable(G);