Properties

Label 40T43
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{20}.C_4$

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magma: G := TransitiveGroup(40, 43);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $43$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{20}.C_4$
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:  (1,21,3,23,2,22,4,24)(5,36,39,10,6,35,40,9)(7,34,38,11,8,33,37,12)(13,18,32,25,14,17,31,26)(15,20,29,27,16,19,30,28), (1,39,9,16,2,40,10,15)(3,37,11,13,4,38,12,14)(5,25,7,28,6,26,8,27)(17,31,35,22,18,32,36,21)(19,30,34,24,20,29,33,23)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$16$:  $C_8:C_2$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: $F_5$

Degree 8: $C_8:C_2$

Degree 10: $F_5$

Degree 20: 20T9

Low degree siblings

There are no siblings 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 $ $10$ $2$ $( 5,37)( 6,38)( 7,39)( 8,40)( 9,34)(10,33)(11,36)(12,35)(13,29)(14,30)(15,31) (16,32)(17,27)(18,28)(19,26)(20,25)(21,22)(23,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,32,30,31)(33,35,34,36)(37,39,38,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 3, 2, 4)( 5,39, 6,40)( 7,38, 8,37)( 9,36,10,35)(11,33,12,34)(13,32,14,31) (15,29,16,30)(17,26,18,25)(19,28,20,27)(21,23,22,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $5$ $4$ $( 1, 4, 2, 3)( 5,40, 6,39)( 7,37, 8,38)( 9,35,10,36)(11,34,12,33)(13,31,14,32) (15,30,16,29)(17,25,18,26)(19,27,20,28)(21,24,22,23)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 5,18,13, 2, 6,17,14)( 3, 8,20,15, 4, 7,19,16)( 9,29,11,31,10,30,12,32) (21,25,37,34,22,26,38,33)(23,27,40,36,24,28,39,35)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 5,34,29, 2, 6,33,30)( 3, 8,36,31, 4, 7,35,32)( 9,22,25,15,10,21,26,16) (11,24,27,14,12,23,28,13)(17,40,20,37,18,39,19,38)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 7,18,16, 2, 8,17,15)( 3, 5,20,13, 4, 6,19,14)( 9,32,11,29,10,31,12,30) (21,27,37,36,22,28,38,35)(23,26,40,33,24,25,39,34)$
$ 8, 8, 8, 8, 8 $ $10$ $8$ $( 1, 7,34,32, 2, 8,33,31)( 3, 5,36,29, 4, 6,35,30)( 9,23,25,13,10,24,26,14) (11,22,27,15,12,21,28,16)(17,37,20,39,18,38,19,40)$
$ 20, 20 $ $4$ $20$ $( 1, 9,20,28,36, 4,12,18,26,34, 2,10,19,27,35, 3,11,17,25,33)( 5,13,23,29,38, 8,15,22,31,39, 6,14,24,30,37, 7,16,21,32,40)$
$ 20, 20 $ $4$ $20$ $( 1,10,20,27,36, 3,12,17,26,33, 2, 9,19,28,35, 4,11,18,25,34)( 5,14,23,30,38, 7,15,21,31,40, 6,13,24,29,37, 8,16,22,32,39)$
$ 5, 5, 5, 5, 5, 5, 5, 5 $ $4$ $5$ $( 1,11,19,26,36)( 2,12,20,25,35)( 3,10,18,28,33)( 4, 9,17,27,34) ( 5,16,24,31,38)( 6,15,23,32,37)( 7,14,22,29,40)( 8,13,21,30,39)$
$ 10, 10, 10, 10 $ $4$ $10$ $( 1,12,19,25,36, 2,11,20,26,35)( 3, 9,18,27,33, 4,10,17,28,34)( 5,15,24,32,38, 6,16,23,31,37)( 7,13,22,30,40, 8,14,21,29,39)$

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:  80.29
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 4a 4b 4c 8a 8b 8c 8d 20a 20b 5a 10a
     2P 1a 1a 1a 2b 2b 2b 4b 4c 4b 4c 10a 10a 5a  5a
     3P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20a 20b 5a 10a
     5P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d  4a  4a 1a  2b
     7P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20a 20b 5a 10a
    11P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20b 20a 5a 10a
    13P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 20b 20a 5a 10a
    17P 1a 2a 2b 4a 4b 4c 8a 8b 8c 8d 20b 20a 5a 10a
    19P 1a 2a 2b 4a 4c 4b 8d 8c 8b 8a 20b 20a 5a 10a

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

A = -2*E(4)
  = -2*Sqrt(-1) = -2i
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(20)-E(20)^9+E(20)^13+E(20)^17
  = -Sqrt(-5) = -i5

magma: CharacterTable(G);