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Properties

Label 40T48
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:D_8$

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Show commands: Magma

magma: G := TransitiveGroup(40, 48);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $48$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:D_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,7,19,11,2,13,8,20,12)(3,15,5,17,10)(4,16,6,18,9)(21,33,27,37,32,24,36,25,39,29)(22,34,28,38,31,23,35,26,40,30), (1,33,3,36,2,34,4,35)(5,32,8,30,6,31,7,29)(9,28,11,25,10,27,12,26)(13,24,15,21,14,23,16,22)(17,39,19,38,18,40,20,37)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$10$:  $D_{5}$
$16$:  $D_{8}$
$20$:  $D_{10}$
$40$:  20T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $D_{5}$

Degree 8: $D_{8}$

Degree 10: $D_5$

Degree 20: 20T11

Low degree siblings

40T32

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

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:  80.15
magma: IdentifyGroup(G);
 
Character table:    
      2  4  2  4  3   2   2   3   2   3   2   2   2   3   3  3  2  3
      5  1  1  1  1   1   1   1   1   1   1   1   1   1   1  .  .  .

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

X.1      1  1  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 -1  1 -1
X.3      1 -1  1  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 -1 -1 -1
X.5      2  .  2 -2   .  -2   2   .   2  -2   .   .   2   2  .  .  .
X.6      2 -2  2  2   A  -A  -A   A  -A -*A  *A  *A -*A -*A  .  .  .
X.7      2 -2  2  2  *A -*A -*A  *A -*A  -A   A   A  -A  -A  .  .  .
X.8      2  . -2  .   .   .   2   .  -2   .   .   .   2  -2  E  . -E
X.9      2  . -2  .   .   .   2   .  -2   .   .   .   2  -2 -E  .  E
X.10     2  .  2 -2   B   A  -A  -B  -A  *A   C  -C -*A -*A  .  .  .
X.11     2  .  2 -2   C  *A -*A  -C -*A   A  -B   B  -A  -A  .  .  .
X.12     2  .  2 -2  -C  *A -*A   C -*A   A   B  -B  -A  -A  .  .  .
X.13     2  .  2 -2  -B   A  -A   B  -A  *A  -C   C -*A -*A  .  .  .
X.14     2  2  2  2 -*A -*A -*A -*A -*A  -A  -A  -A  -A  -A  .  .  .
X.15     2  2  2  2  -A  -A  -A  -A  -A -*A -*A -*A -*A -*A  .  .  .
X.16     4  . -4  .   .   .   D   .  -D   .   .   .  *D -*D  .  .  .
X.17     4  . -4  .   .   .  *D   . -*D   .   .   .   D  -D  .  .  .

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
B = -E(5)+E(5)^4
C = -E(5)^2+E(5)^3
D = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
E = -E(8)+E(8)^3
  = -Sqrt(2) = -r2

magma: CharacterTable(G);