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Properties

Label 40T45
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, 45);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{10}:C_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,39,27,31)(2,40,28,32)(3,37,25,29)(4,38,26,30)(5,12,21,18)(6,11,22,17)(7,9,24,20)(8,10,23,19)(13,34,15,36)(14,33,16,35), (1,22,4,24)(2,21,3,23)(5,11,39,34)(6,12,40,33)(7,10,37,36)(8,9,38,35)(13,27,31,20)(14,28,32,19)(15,25,29,18)(16,26,30,17)
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: 20T9, 20T22 x 2

Low degree siblings

20T19 x 2, 20T22 x 2, 40T26, 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, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $10$ $2$ $( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,25)(20,26)(21,22)(23,24)$
$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,32)(30,31)(33,35)(34,36)(37,40)(38,39)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 3)( 2, 4)( 5,40)( 6,39)( 7,38)( 8,37)( 9,36)(10,35)(11,33)(12,34)(13,32) (14,31)(15,30)(16,29)(17,25)(18,26)(19,27)(20,28)(21,24)(22,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 4)( 2, 3)( 5,39)( 6,40)( 7,37)( 8,38)( 9,35)(10,36)(11,34)(12,33)(13,31) (14,32)(15,29)(16,30)(17,26)(18,25)(19,28)(20,27)(21,23)(22,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 5,18,14)( 2, 6,17,13)( 3, 7,20,16)( 4, 8,19,15)( 9,29,11,32)(10,30,12,31) (21,26,37,33)(22,25,38,34)(23,27,40,36)(24,28,39,35)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 5,33,30)( 2, 6,34,29)( 3, 7,35,32)( 4, 8,36,31)( 9,22,26,16)(10,21,25,15) (11,24,27,13)(12,23,28,14)(17,40,19,38)(18,39,20,37)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 7,17,15)( 2, 8,18,16)( 3, 5,19,13)( 4, 6,20,14)( 9,31,12,29)(10,32,11,30) (21,27,38,35)(22,28,37,36)(23,26,39,34)(24,25,40,33)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $10$ $4$ $( 1, 7,34,31)( 2, 8,33,32)( 3, 5,36,29)( 4, 6,35,30)( 9,23,25,13)(10,24,26,14) (11,21,28,16)(12,22,27,15)(17,38,20,39)(18,37,19,40)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1, 9,19,28,35, 4,11,18,25,34)( 2,10,20,27,36, 3,12,17,26,33)( 5,13,24,30,38, 7,15,21,32,40)( 6,14,23,29,37, 8,16,22,31,39)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,10,19,27,35, 3,11,17,25,33)( 2, 9,20,28,36, 4,12,18,26,34)( 5,14,24,29,38, 8,15,22,32,39)( 6,13,23,30,37, 7,16,21,31,40)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,11,19,25,35)( 2,12,20,26,36)( 3,10,17,27,33)( 4, 9,18,28,34) ( 5,15,24,32,38)( 6,16,23,31,37)( 7,13,21,30,40)( 8,14,22,29,39)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,12,19,26,35, 2,11,20,25,36)( 3, 9,17,28,33, 4,10,18,27,34)( 5,16,24,31,38, 6,15,23,32,37)( 7,14,21,29,40, 8,13,22,30,39)$

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:  80.34
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);