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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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Copy content magma:G := TransitiveGroup(40, 21);
 

Group invariants

Abstract group:  $D_4:D_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $80=2^{4} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{40}$ $1$ $1$ $0$ $()$
2A $2^{20}$ $1$ $2$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2B $2^{10},1^{20}$ $2$ $2$ $10$ $( 1, 2)( 3, 4)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)(33,34)(35,36)$
2C $2^{20}$ $2$ $2$ $20$ $( 1,24)( 2,23)( 3,22)( 4,21)( 5,26)( 6,25)( 7,28)( 8,27)( 9,30)(10,29)(11,31)(12,32)(13,34)(14,33)(15,35)(16,36)(17,39)(18,40)(19,38)(20,37)$
2D $2^{20}$ $10$ $2$ $20$ $( 1,22)( 2,21)( 3,23)( 4,24)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)(25,37)(26,38)(27,40)(28,39)(29,36)(30,35)(31,33)(32,34)$
4A $4^{10}$ $2$ $4$ $30$ $( 1,24, 2,23)( 3,22, 4,21)( 5,25, 6,26)( 7,27, 8,28)( 9,30,10,29)(11,31,12,32)(13,33,14,34)(15,36,16,35)(17,39,18,40)(19,38,20,37)$
4B1 $4^{10}$ $5$ $4$ $30$ $( 1, 9, 2,10)( 3,11, 4,12)( 5, 7, 6, 8)(13,38,14,37)(15,40,16,39)(17,35,18,36)(19,33,20,34)(21,32,22,31)(23,29,24,30)(25,27,26,28)$
4B-1 $4^{10}$ $5$ $4$ $30$ $( 1,10, 2, 9)( 3,12, 4,11)( 5, 8, 6, 7)(13,37,14,38)(15,39,16,40)(17,36,18,35)(19,34,20,33)(21,31,22,32)(23,30,24,29)(25,28,26,27)$
4C $4^{10}$ $10$ $4$ $30$ $( 1,13, 2,14)( 3,15, 4,16)( 5,11, 6,12)( 7, 9, 8,10)(17,38,18,37)(19,40,20,39)(21,36,22,35)(23,33,24,34)(25,32,26,31)(27,29,28,30)$
4D $4^{10}$ $10$ $4$ $30$ $( 1,18, 2,17)( 3,20, 4,19)( 5,15, 6,16)( 7,13, 8,14)( 9,11,10,12)(21,37,22,38)(23,40,24,39)(25,35,26,36)(27,34,28,33)(29,31,30,32)$
5A1 $5^{8}$ $2$ $5$ $32$ $( 1,36,27,19,11)( 2,35,28,20,12)( 3,34,25,18,10)( 4,33,26,17, 9)( 5,39,30,21,14)( 6,40,29,22,13)( 7,37,32,23,15)( 8,38,31,24,16)$
5A2 $5^{8}$ $2$ $5$ $32$ $( 1,27,11,36,19)( 2,28,12,35,20)( 3,25,10,34,18)( 4,26, 9,33,17)( 5,30,14,39,21)( 6,29,13,40,22)( 7,32,15,37,23)( 8,31,16,38,24)$
10A1 $10^{4}$ $2$ $10$ $36$ $( 1,35,27,20,11, 2,36,28,19,12)( 3,33,25,17,10, 4,34,26,18, 9)( 5,40,30,22,14, 6,39,29,21,13)( 7,38,32,24,15, 8,37,31,23,16)$
10A3 $10^{4}$ $2$ $10$ $36$ $( 1,20,36,12,27, 2,19,35,11,28)( 3,17,34, 9,25, 4,18,33,10,26)( 5,22,39,13,30, 6,21,40,14,29)( 7,24,37,16,32, 8,23,38,15,31)$
10B1 $10^{2},5^{4}$ $4$ $10$ $34$ $( 1,20,36,12,27, 2,19,35,11,28)( 3,17,34, 9,25, 4,18,33,10,26)( 5,21,39,14,30)( 6,22,40,13,29)( 7,23,37,15,32)( 8,24,38,16,31)$
10B3 $10^{2},5^{4}$ $4$ $10$ $34$ $( 1,12,19,28,36, 2,11,20,27,35)( 3, 9,18,26,34, 4,10,17,25,33)( 5,14,21,30,39)( 6,13,22,29,40)( 7,15,23,32,37)( 8,16,24,31,38)$
10C1 $10^{4}$ $4$ $10$ $36$ $( 1,16,27,38,11,24,36, 8,19,31)( 2,15,28,37,12,23,35, 7,20,32)( 3,13,25,40,10,22,34, 6,18,29)( 4,14,26,39, 9,21,33, 5,17,30)$
10C3 $10^{4}$ $4$ $10$ $36$ $( 1, 8,11,16,19,24,27,31,36,38)( 2, 7,12,15,20,23,28,32,35,37)( 3, 6,10,13,18,22,25,29,34,40)( 4, 5, 9,14,17,21,26,30,33,39)$
20A1 $20^{2}$ $4$ $20$ $38$ $( 1,38,35,32,27,24,20,15,11, 8, 2,37,36,31,28,23,19,16,12, 7)( 3,40,33,30,25,22,17,14,10, 6, 4,39,34,29,26,21,18,13, 9, 5)$
20A3 $20^{2}$ $4$ $20$ $38$ $( 1,31,20, 7,36,24,12,37,27,16, 2,32,19, 8,35,23,11,38,28,15)( 3,29,17, 5,34,22, 9,39,25,13, 4,30,18, 6,33,21,10,40,26,14)$

Malle's constant $a(G)$:     $1/10$

Copy content magma:ConjugacyClasses(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 5A1 5A2 5A2 5A1 10A1 10A3
5 P 1A 2A 2B 2C 2D 4A 4B1 4B-1 4C 4D 1A 1A 2A 2A 2B 2B 2C 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 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 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 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 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 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −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 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −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 ζ5−2+ζ52 −ζ5−2−ζ52 ζ5−1+ζ5 −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 ζ5−1+ζ5 −ζ5−1−ζ5 ζ5−2+ζ52 −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 −ζ5−2−ζ52 ζ5−2+ζ52 −ζ5−1−ζ5 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 −ζ5−1−ζ5 ζ5−1+ζ5 −ζ5−2−ζ52 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

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed