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Properties

Label 40T39
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4\times D_5$

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Show commands: Magma

magma: G := TransitiveGroup(40, 39);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4\times D_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,4)(2,3)(5,40)(6,39)(7,37)(8,38)(9,36)(10,35)(11,33)(12,34)(13,32)(14,31)(15,30)(16,29)(17,26)(18,25)(19,27)(20,28)(21,23)(22,24), (1,15,25,39,12,23,35,7,20,31,2,16,26,40,11,24,36,8,19,32)(3,13,28,38,10,22,34,5,18,29,4,14,27,37,9,21,33,6,17,30), (1,31)(2,32)(3,29)(4,30)(5,28)(6,27)(7,25)(8,26)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,37)(34,38)(35,39)(36,40)
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
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T8

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 5: $D_{5}$

Degree 8: $D_4\times C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8, 20T21 x 2

Low degree siblings

20T21 x 4, 40T22 x 2, 40T39, 40T40 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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);
 
Nilpotency class:   not nilpotent
Label:  80.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3 20A1 20A3
Size 1 1 2 2 5 5 10 10 2 10 2 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 5A2 5A1 5A1 5A2 5A1 5A2 5A1 5A2 10A1 10A3
5 P 1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 1A 1A 2A 2A 2B 2B 2C 2C 4A 4A
Type
80.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.39.1b R 1 1 −1 −1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1c R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.39.1d R 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1e R 1 1 −1 1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1
80.39.1f R 1 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1g R 1 1 1 −1 1 1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
80.39.1h R 1 1 1 1 −1 −1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1 1
80.39.2a R 2 −2 0 0 −2 2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2b R 2 −2 0 0 2 −2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.39.2c1 R 2 2 2 2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2c2 R 2 2 2 2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2d1 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.39.2d2 R 2 2 −2 −2 0 0 0 0 2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.39.2e1 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.2e2 R 2 2 −2 2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.2f1 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.39.2f2 R 2 2 2 −2 0 0 0 0 −2 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.39.4a1 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0
80.39.4a2 R 4 −4 0 0 0 0 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0

magma: CharacterTable(G);