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Properties

Label 40T10
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times D_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(40, 10);
 

Group action invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 5: $D_{5}$

Degree 8: $C_2^3$

Degree 10: $D_5$, $D_{10}$ x 6

Degree 20: 20T4 x 3, 20T8 x 4

Low degree siblings

20T8 x 4

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

Malle's constant $a(G)$:     $1/20$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  40.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 5A1 5A2 10A1 10A3 10B1 10B3 10C1 10C3
Size 1 1 1 1 5 5 5 5 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 5A2 5A1 5A2 5A1 5A2 5A1 5A2 5A1
5 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 2B 2B 2C 2A 2A 2C
Type
40.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
40.13.1b R 1 −1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 −1 1 1
40.13.1c R 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
40.13.1d R 1 −1 1 −1 −1 1 −1 1 1 1 −1 −1 1 1 −1 −1
40.13.1e R 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1
40.13.1f R 1 1 −1 −1 −1 1 1 −1 1 1 1 1 −1 −1 −1 −1
40.13.1g R 1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
40.13.1h R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1
40.13.2a1 R 2 2 2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
40.13.2a2 R 2 2 2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
40.13.2b1 R 2 −2 −2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5
40.13.2b2 R 2 −2 −2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52
40.13.2c1 R 2 −2 2 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
40.13.2c2 R 2 −2 2 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
40.13.2d1 R 2 2 −2 −2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5
40.13.2d2 R 2 2 −2 −2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52

magma: CharacterTable(G);