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Properties

Label 40T26
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{10}:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(40, 26);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{10}:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $20$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,30,33,8)(2,29,34,7)(3,31,36,5)(4,32,35,6)(9,14,28,23)(10,13,27,24)(11,16,26,22)(12,15,25,21)(17,39,20,38)(18,40,19,37), (1,11,19,27,36,2,12,20,28,35)(3,10,18,26,33,4,9,17,25,34)(5,15,23,30,37)(6,16,24,29,38)(7,13,22,32,39)(8,14,21,31,40)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 5: $F_5$

Degree 8: $C_2^2:C_4$

Degree 10: $F_5$

Degree 20: 20T5, 20T22 x 2

Low degree siblings

20T19 x 2, 20T22 x 2, 40T45, 40T55 x 2

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

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:  80.34
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A1 4A-1 4B1 4B-1 5A 10A 10B1 10B3
Size 1 1 2 5 5 10 10 10 10 10 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2D 2D 2C 2C 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 4B-1 4B1 4A-1 4A1 1A 2A 2B 2B
Type
80.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.34.1b R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1
80.34.1c R 1 1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1
80.34.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
80.34.1e1 C 1 1 −1 −1 −1 1 −i i −i i 1 1 −1 −1
80.34.1e2 C 1 1 −1 −1 −1 1 i −i i −i 1 1 −1 −1
80.34.1f1 C 1 1 1 −1 −1 −1 −i i i −i 1 1 1 1
80.34.1f2 C 1 1 1 −1 −1 −1 i −i −i i 1 1 1 1
80.34.2a R 2 −2 0 −2 2 0 0 0 0 0 2 −2 0 0
80.34.2b R 2 −2 0 2 −2 0 0 0 0 0 2 −2 0 0
80.34.4a R 4 4 4 0 0 0 0 0 0 0 −1 −1 −1 −1
80.34.4b R 4 4 −4 0 0 0 0 0 0 0 −1 −1 1 1
80.34.4c1 R 4 −4 0 0 0 0 0 0 0 0 −1 1 2ζ5−2+1+2ζ52 −2ζ5−2−1−2ζ52
80.34.4c2 R 4 −4 0 0 0 0 0 0 0 0 −1 1 −2ζ5−2−1−2ζ52 2ζ5−2+1+2ζ52

magma: CharacterTable(G);