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Properties

Label 40T49
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:\SD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(40, 49);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:\SD_{16}$
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:  (1,33,3,35,2,34,4,36)(5,31,8,30,6,32,7,29)(9,27,11,25,10,28,12,26)(13,24,15,22,14,23,16,21)(17,40,19,38,18,39,20,37), (1,15,7,17,11,3,13,5,20,10)(2,16,8,18,12,4,14,6,19,9)(21,36,27,39,32)(22,35,28,40,31)(23,33,26,37,30,24,34,25,38,29)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $QD_{16}$

Degree 10: $D_5$

Degree 20: 20T11

Low degree siblings

There are no siblings 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 $2^{15},1^{10}$ $4$ $2$ $15$ $( 3, 4)( 5, 6)( 9,10)(15,16)(17,18)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)(33,35)(34,36)(37,40)(38,39)$
4A $4^{10}$ $2$ $4$ $30$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,28,26,27)(29,31,30,32)(33,35,34,36)(37,40,38,39)$
4B $4^{10}$ $20$ $4$ $30$ $( 1,35, 2,36)( 3,33, 4,34)( 5,29, 6,30)( 7,31, 8,32)( 9,26,10,25)(11,28,12,27)(13,22,14,21)(15,24,16,23)(17,37,18,38)(19,39,20,40)$
5A1 $5^{8}$ $2$ $5$ $32$ $( 1,20,13,11, 7)( 2,19,14,12, 8)( 3,17,15,10, 5)( 4,18,16, 9, 6)(21,39,36,32,27)(22,40,35,31,28)(23,38,34,30,26)(24,37,33,29,25)$
5A2 $5^{8}$ $2$ $5$ $32$ $( 1,11,20, 7,13)( 2,12,19, 8,14)( 3,10,17, 5,15)( 4, 9,18, 6,16)(21,32,39,27,36)(22,31,40,28,35)(23,30,38,26,34)(24,29,37,25,33)$
8A1 $8^{5}$ $10$ $8$ $35$ $( 1,33, 3,35, 2,34, 4,36)( 5,31, 8,30, 6,32, 7,29)( 9,27,11,25,10,28,12,26)(13,24,15,22,14,23,16,21)(17,40,19,38,18,39,20,37)$
8A-1 $8^{5}$ $10$ $8$ $35$ $( 1,34, 3,36, 2,33, 4,35)( 5,32, 8,29, 6,31, 7,30)( 9,28,11,26,10,27,12,25)(13,23,15,21,14,24,16,22)(17,39,19,37,18,40,20,38)$
10A1 $10^{4}$ $2$ $10$ $36$ $( 1,19,13,12, 7, 2,20,14,11, 8)( 3,18,15, 9, 5, 4,17,16,10, 6)(21,40,36,31,27,22,39,35,32,28)(23,37,34,29,26,24,38,33,30,25)$
10A3 $10^{4}$ $2$ $10$ $36$ $( 1,12,20, 8,13, 2,11,19, 7,14)( 3, 9,17, 6,15, 4,10,18, 5,16)(21,31,39,28,36,22,32,40,27,35)(23,29,38,25,34,24,30,37,26,33)$
10B1 $10^{3},5^{2}$ $4$ $10$ $35$ $( 1,11,20, 7,13)( 2,12,19, 8,14)( 3, 9,17, 6,15, 4,10,18, 5,16)(21,30,39,26,36,23,32,38,27,34)(22,29,40,25,35,24,31,37,28,33)$
10B-1 $10^{3},5^{2}$ $4$ $10$ $35$ $( 1,13, 7,20,11)( 2,14, 8,19,12)( 3,16, 5,18,10, 4,15, 6,17, 9)(21,34,27,38,32,23,36,26,39,30)(22,33,28,37,31,24,35,25,40,29)$
10B3 $10^{3},5^{2}$ $4$ $10$ $35$ $( 1, 7,11,13,20)( 2, 8,12,14,19)( 3, 6,10,16,17, 4, 5, 9,15,18)(21,26,32,34,39,23,27,30,36,38)(22,25,31,33,40,24,28,29,35,37)$
10B-3 $10^{3},5^{2}$ $4$ $10$ $35$ $( 1,20,13,11, 7)( 2,19,14,12, 8)( 3,18,15, 9, 5, 4,17,16,10, 6)(21,38,36,30,27,23,39,34,32,26)(22,37,35,29,28,24,40,33,31,25)$
20A1 $20^{2}$ $4$ $20$ $38$ $( 1,10,19, 6,13, 3,12,18, 7,15, 2, 9,20, 5,14, 4,11,17, 8,16)(21,29,40,26,36,24,31,38,27,33,22,30,39,25,35,23,32,37,28,34)$
20A3 $20^{2}$ $4$ $20$ $38$ $( 1,17,14, 9, 7, 3,19,16,11, 5, 2,18,13,10, 8, 4,20,15,12, 6)(21,37,35,30,27,24,40,34,32,25,22,38,36,29,28,23,39,33,31,26)$

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:  80.16
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A 4B 5A1 5A2 8A1 8A-1 10A1 10A3 10B1 10B-1 10B3 10B-3 20A1 20A3
Size 1 1 4 2 20 2 2 10 10 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 2A 2A 5A2 5A1 4A 4A 5A2 5A1 5A1 5A1 5A2 5A2 10A1 10A3
5 P 1A 2A 2B 4A 4B 1A 1A 8A-1 8A1 2A 2A 2B 2B 2B 2B 4A 4A
Type
80.16.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.16.1b R 1 1 −1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.16.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
80.16.1d R 1 1 1 1 −1 1 1 −1 −1 1 1 1 1 1 1 1 1
80.16.2a R 2 2 0 −2 0 2 2 0 0 2 2 0 0 0 0 −2 −2
80.16.2b1 R 2 2 2 2 0 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
80.16.2b2 R 2 2 2 2 0 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
80.16.2c1 C 2 −2 0 0 0 2 2 −ζ8−ζ83 ζ8+ζ83 −2 −2 0 0 0 0 0 0
80.16.2c2 C 2 −2 0 0 0 2 2 ζ8+ζ83 −ζ8−ζ83 −2 −2 0 0 0 0 0 0
80.16.2d1 R 2 2 −2 2 0 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
80.16.2d2 R 2 2 −2 2 0 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
80.16.2e1 C 2 2 0 −2 0 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
80.16.2e2 C 2 2 0 −2 0 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
80.16.2e3 C 2 2 0 −2 0 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
80.16.2e4 C 2 2 0 −2 0 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
80.16.4a1 S 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
80.16.4a2 S 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);