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Properties

Label 40T35
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_4:D_5$

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Show commands: Magma

magma: G := TransitiveGroup(40, 35);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:D_5$
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:  (1,5,2,6)(3,7,4,8)(9,39,10,40)(11,37,12,38)(13,36,14,35)(15,34,16,33)(17,32,18,31)(19,30,20,29)(21,25,22,26)(23,28,24,27), (1,12)(2,11)(3,9)(4,10)(7,8)(13,38)(14,37)(15,40)(16,39)(17,33)(18,34)(19,36)(20,35)(21,30)(22,29)(23,31)(24,32)(27,28), (1,33,2,34)(3,35,4,36)(5,31,6,32)(7,29,8,30)(9,26,10,25)(11,27,12,28)(13,24,14,23)(15,21,16,22)(17,20,18,19)(37,40,38,39)
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
$8$:  $C_2^3$
$10$:  $D_{5}$
$16$:  $Q_8:C_2$
$20$:  $D_{10}$ 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: $Q_8:C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8

Low degree siblings

40T21 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 $ $10$ $2$ $( 3, 4)( 5,38)( 6,37)( 7,39)( 8,40)( 9,34)(10,33)(11,36)(12,35)(13,29)(14,30) (15,32)(16,31)(17,28)(18,27)(19,26)(20,25)(23,24)$
$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 3, 2, 4)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,33,12,34)(13,32,14,31) (15,29,16,30)(17,26,18,25)(19,28,20,27)(21,24,22,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,39,10,40)(11,37,12,38)(13,36,14,35)(15,34,16,33) (17,32,18,31)(19,30,20,29)(21,25,22,26)(23,28,24,27)$
$ 20, 20 $ $4$ $20$ $( 1, 5,11,14,20,22,26,30,35,38, 2, 6,12,13,19,21,25,29,36,37)( 3, 8, 9,16,17, 24,28,32,34,39, 4, 7,10,15,18,23,27,31,33,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,40,10,39)(11,38,12,37)(13,35,14,36)(15,33,16,34) (17,31,18,32)(19,29,20,30)(21,26,22,25)(23,27,24,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,37,10,38)(11,39,12,40)(13,33,14,34)(15,35,16,36) (17,29,18,30)(19,31,20,32)(21,27,22,28)(23,26,24,25)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1, 7,12,16,20,23,25,32,35,40)( 2, 8,11,15,19,24,26,31,36,39)( 3, 6,10,14,17, 21,27,30,34,37)( 4, 5, 9,13,18,22,28,29,33,38)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1, 9,20,28,35, 4,12,18,25,33)( 2,10,19,27,36, 3,11,17,26,34)( 5,15,22,31,38, 8,13,24,29,39)( 6,16,21,32,37, 7,14,23,30,40)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,11,20,26,35, 2,12,19,25,36)( 3, 9,17,28,34, 4,10,18,27,33)( 5,14,22,30,38, 6,13,21,29,37)( 7,15,23,31,40, 8,16,24,32,39)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,12,20,25,35)( 2,11,19,26,36)( 3,10,17,27,34)( 4, 9,18,28,33) ( 5,13,22,29,38)( 6,14,21,30,37)( 7,16,23,32,40)( 8,15,24,31,39)$
$ 20, 20 $ $4$ $20$ $( 1,13,26,37,12,22,36, 6,20,29, 2,14,25,38,11,21,35, 5,19,30)( 3,15,28,40,10, 24,33, 7,17,31, 4,16,27,39, 9,23,34, 8,18,32)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,15,25,39,12,24,35, 8,20,31)( 2,16,26,40,11,23,36, 7,19,32)( 3,13,27,38,10, 22,34, 5,17,29)( 4,14,28,37, 9,21,33, 6,18,30)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,17,35,10,25, 3,20,34,12,27)( 2,18,36, 9,26, 4,19,33,11,28)( 5,23,38,16,29, 7,22,40,13,32)( 6,24,37,15,30, 8,21,39,14,31)$
$ 10, 10, 10, 10 $ $2$ $10$ $( 1,19,35,11,25, 2,20,36,12,26)( 3,18,34, 9,27, 4,17,33,10,28)( 5,21,38,14,29, 6,22,37,13,30)( 7,24,40,15,32, 8,23,39,16,31)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $2$ $5$ $( 1,20,35,12,25)( 2,19,36,11,26)( 3,17,34,10,27)( 4,18,33, 9,28) ( 5,22,38,13,29)( 6,21,37,14,30)( 7,23,40,16,32)( 8,24,39,15,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 2,22)( 3,23, 4,24)( 5,25, 6,26)( 7,28, 8,27)( 9,31,10,32)(11,29,12,30) (13,35,14,36)(15,34,16,33)(17,40,18,39)(19,38,20,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,23)( 2,24)( 3,21)( 4,22)( 5,28)( 6,27)( 7,25)( 8,26)( 9,29)(10,30)(11,31) (12,32)(13,33)(14,34)(15,36)(16,35)(17,37)(18,38)(19,39)(20,40)$

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:  80.40
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A 4B1 4B-1 4C 4D 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3 20A1 20A3
Size 1 1 2 2 10 2 5 5 10 10 2 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 5A2 5A1 5A2 5A1 5A2 5A2 5A1 5A1 10A1 10A3
5 P 1A 2A 2B 2C 2D 4A 4B1 4B-1 4C 4D 1A 1A 2A 2A 2B 2C 2B 2C 4A 4A
Type
80.40.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.40.1b R 1 1 −1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.40.1c R 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
80.40.1d R 1 1 −1 1 −1 −1 1 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1
80.40.1e R 1 1 −1 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
80.40.1f R 1 1 1 −1 −1 −1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1
80.40.1g R 1 1 1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1
80.40.1h R 1 1 1 1 −1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1
80.40.2a1 R 2 2 2 2 0 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.40.2a2 R 2 2 2 2 0 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.40.2b1 C 2 −2 0 0 0 0 −2i 2i 0 0 2 2 −2 −2 0 0 0 0 0 0
80.40.2b2 C 2 −2 0 0 0 0 2i −2i 0 0 2 2 −2 −2 0 0 0 0 0 0
80.40.2c1 R 2 2 −2 −2 0 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.40.2c2 R 2 2 −2 −2 0 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.40.2d1 R 2 2 −2 2 0 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.40.2d2 R 2 2 −2 2 0 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.40.2e1 R 2 2 2 −2 0 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.40.2e2 R 2 2 2 −2 0 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.40.4a1 S 4 −4 0 0 0 0 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0
80.40.4a2 S 4 −4 0 0 0 0 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0

magma: CharacterTable(G);