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Properties

Label 40T11
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(40, 11);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $11$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:D_4$
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:  (1,14)(2,13)(3,15)(4,16)(5,12)(6,11)(7,9)(8,10)(17,38)(18,37)(19,40)(20,39)(21,35)(22,36)(23,33)(24,34)(25,32)(26,31)(27,30)(28,29), (1,17,2,18)(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

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 5: $D_{5}$

Degree 8: $D_4$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4, 20T7, 20T11

Low degree siblings

20T7, 20T11

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

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:  40.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 10A1 10A3 10B1 10B-1 10B3 10B-3
Size 1 1 2 10 10 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A2 5A1 5A2 5A1 5A1 5A2
5 P 1A 2A 2B 2C 4A 1A 1A 2A 2A 2B 2B 2B 2B
Type
40.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.8.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
40.8.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.8.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.8.2a R 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.8.2b1 R 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
40.8.2b2 R 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
40.8.2c1 R 2 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52
40.8.2c2 R 2 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5
40.8.2d1 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52
40.8.2d2 C 2 −2 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52
40.8.2d3 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52
40.8.2d4 C 2 −2 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52

magma: CharacterTable(G);