↑

Properties

Label 40T34
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $Q_8\times D_5$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(40, 34);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8\times D_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24,2,23)(3,21,4,22)(5,17,6,18)(7,20,8,19)(9,13,10,14)(11,15,12,16)(25,39,26,40)(27,37,28,38)(29,33,30,34)(31,36,32,35), (1,18,2,17)(3,19,4,20)(5,16,6,15)(7,13,8,14)(9,12,10,11)(21,40,22,39)(23,37,24,38)(25,34,26,33)(27,35,28,36)(29,32,30,31), (1,15,26,39,12,23,35,7,19,31,2,16,25,40,11,24,36,8,20,32)(3,14,27,38,9,22,34,6,17,29,4,13,28,37,10,21,33,5,18,30)
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
$8$:  $C_2^3$, $Q_8$ x 2
$10$:  $D_{5}$
$16$:  $Q_8\times C_2$
$20$:  $D_{10}$ 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: $Q_8$

Degree 10: $D_{10}$ x 3

Degree 20: 20T8

Low degree siblings

40T34

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

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:  80.41
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A1 5A2 10A1 10A3 20A1 20A3 20B1 20B3 20C1 20C3
Size 1 1 5 5 2 2 2 10 10 10 2 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 5A2 5A1 5A1 5A2 10A1 10A3 10A1 10A3 10A3 10A1
5 P 1A 2A 2B 2C 4A 4B 4C 4E 4F 4D 1A 1A 2A 2A 4C 4B 4B 4A 4C 4A
Type
80.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.41.1b R 1 1 −1 −1 −1 −1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.41.1c R 1 1 −1 −1 −1 1 −1 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1
80.41.1d R 1 1 −1 −1 1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
80.41.1e R 1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1
80.41.1f R 1 1 1 1 −1 −1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1
80.41.1g R 1 1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 −1 −1 1 1 −1 −1
80.41.1h R 1 1 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
80.41.2a S 2 −2 −2 2 0 0 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.41.2b S 2 −2 2 −2 0 0 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0
80.41.2c1 R 2 2 0 0 2 2 2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.41.2c2 R 2 2 0 0 2 2 2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.41.2d1 R 2 2 0 0 −2 −2 2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5
80.41.2d2 R 2 2 0 0 −2 −2 2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52
80.41.2e1 R 2 2 0 0 −2 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.41.2e2 R 2 2 0 0 −2 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.41.2f1 R 2 2 0 0 2 −2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.41.2f2 R 2 2 0 0 2 −2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
80.41.4a1 S 4 −4 0 0 0 0 0 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0
80.41.4a2 S 4 −4 0 0 0 0 0 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0

magma: CharacterTable(G);