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Properties

Label 40T21
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, 21);
 

Group action invariants

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

Degree 20: 20T4

Low degree siblings

40T21, 40T35

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

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);