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Properties

Label 40T33
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(40, 33);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{10}:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,40,36,31,26,24,19,16,11,8)(2,39,35,32,25,23,20,15,12,7)(3,38,33,30,27,22,18,13,10,5)(4,37,34,29,28,21,17,14,9,6), (1,7,11,15,19,23,26,32,36,39)(2,8,12,16,20,24,25,31,35,40)(3,6,10,14,18,21,27,29,33,37)(4,5,9,13,17,22,28,30,34,38), (1,7,3,6)(2,8,4,5)(9,38,12,40)(10,37,11,39)(13,35,16,34)(14,36,15,33)(17,30,20,31)(18,29,19,32)(21,26,23,27)(22,25,24,28)
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
$8$:  $D_{4}$ x 2, $C_2^3$
$10$:  $D_{5}$
$16$:  $D_4\times C_2$
$20$:  $D_{10}$ x 3
$40$:  20T7 x 2, 20T8

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: $D_4\times C_2$

Degree 10: $D_{10}$ x 3

Degree 20: 20T7 x 2, 20T8

Low degree siblings

40T24 x 2, 40T33

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

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:  80.44
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);