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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);
 
Nilpotency class:  $-1$ (not nilpotent)
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

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 $ $10$ $2$ $( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,35)(12,36)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,25)(20,26)(21,22)(23,24)$
$ 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);
 
Label:  80.34
magma: IdentifyGroup(G);
 
Character table:    
      2  4  3  4  3  4  4  3  3  3  3   2   2  2   2
      5  1  .  1  1  .  .  .  .  .  .   1   1  1   1

        1a 2a 2b 2c 2d 2e 4a 4b 4c 4d 10a 10b 5a 10c
     2P 1a 1a 1a 1a 1a 1a 2e 2d 2d 2e  5a  5a 5a  5a
     3P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10b 10a 5a 10c
     5P 1a 2a 2b 2c 2d 2e 4a 4b 4c 4d  2c  2c 1a  2b
     7P 1a 2a 2b 2c 2d 2e 4d 4c 4b 4a 10b 10a 5a 10c

X.1      1  1  1  1  1  1  1  1  1  1   1   1  1   1
X.2      1 -1  1 -1  1  1 -1  1  1 -1  -1  -1  1   1
X.3      1 -1  1 -1  1  1  1 -1 -1  1  -1  -1  1   1
X.4      1  1  1  1  1  1 -1 -1 -1 -1   1   1  1   1
X.5      1 -1  1  1 -1 -1  A -A  A -A   1   1  1   1
X.6      1 -1  1  1 -1 -1 -A  A -A  A   1   1  1   1
X.7      1  1  1 -1 -1 -1  A  A -A -A  -1  -1  1   1
X.8      1  1  1 -1 -1 -1 -A -A  A  A  -1  -1  1   1
X.9      2  . -2  . -2  2  .  .  .  .   .   .  2  -2
X.10     2  . -2  .  2 -2  .  .  .  .   .   .  2  -2
X.11     4  .  4 -4  .  .  .  .  .  .   1   1 -1  -1
X.12     4  .  4  4  .  .  .  .  .  .  -1  -1 -1  -1
X.13     4  . -4  .  .  .  .  .  .  .   B  -B -1   1
X.14     4  . -4  .  .  .  .  .  .  .  -B   B -1   1

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(5)+E(5)^2+E(5)^3-E(5)^4
  = -Sqrt(5) = -r5

magma: CharacterTable(G);