↑

Properties

Label 40T48
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_8$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(40, 48);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:D_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,7,19,11,2,13,8,20,12)(3,15,5,17,10)(4,16,6,18,9)(21,33,27,37,32,24,36,25,39,29)(22,34,28,38,31,23,35,26,40,30), (1,33,3,36,2,34,4,35)(5,32,8,30,6,31,7,29)(9,28,11,25,10,27,12,26)(13,24,15,21,14,23,16,22)(17,39,19,38,18,40,20,37)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$16$:  $D_{8}$
$20$:  $D_{10}$
$40$:  20T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

Degree 10: $D_5$

Degree 20: 20T11

Low degree siblings

40T32

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

Malle's constant $a(G)$:     $1/15$

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:  80.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 8A1 8A3 10A1 10A3 10B1 10B-1 10B3 10B-3 20A1 20A3
Size 1 1 4 20 2 2 2 10 10 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 2A 5A2 5A1 4A 4A 5A2 5A1 5A1 5A2 5A2 5A1 10A1 10A3
5 P 1A 2A 2B 2C 4A 1A 1A 8A3 8A1 2A 2A 2B 2B 2B 2B 4A 4A
Type
80.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.15.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.15.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
80.15.1d R 1 1 1 −1 1 1 1 −1 −1 1 1 1 1 1 1 1 1
80.15.2a R 2 2 0 0 −2 2 2 0 0 2 2 0 0 0 0 −2 −2
80.15.2b1 R 2 2 2 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
80.15.2b2 R 2 2 2 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
80.15.2c1 R 2 −2 0 0 0 2 2 −ζ8−1−ζ8 ζ8−1+ζ8 −2 −2 0 0 0 0 0 0
80.15.2c2 R 2 −2 0 0 0 2 2 ζ8−1+ζ8 −ζ8−1−ζ8 −2 −2 0 0 0 0 0 0
80.15.2d1 R 2 2 −2 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52
80.15.2d2 R 2 2 −2 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5
80.15.2e1 C 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
80.15.2e2 C 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
80.15.2e3 C 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.15.2e4 C 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.15.4a1 R 4 −4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0
80.15.4a2 R 4 −4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0

magma: CharacterTable(G);