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Properties

Label 40T40
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, 40);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
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);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,32)(2,31)(3,30)(4,29)(5,27)(6,28)(7,26)(8,25)(9,21)(10,22)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)(33,38)(34,37)(35,40)(36,39), (1,35)(2,36)(3,34)(4,33)(5,30)(6,29)(7,32)(8,31)(9,28)(10,27)(11,25)(12,26)(13,21)(14,22)(15,24)(16,23), (1,27,12,34,19,3,26,10,35,17)(2,28,11,33,20,4,25,9,36,18)(5,31,14,39,22,8,30,15,37,24)(6,32,13,40,21,7,29,16,38,23)
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: $D_4$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8, 20T21 x 2

Low degree siblings

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

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

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);
 
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);