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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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Show commands: Magma

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

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

1A 2A 2B 4A 4B1 4B-1 5A 8A1 8A-1 8B1 8B-1 10A 20A1 20A-1
Size 1 1 10 2 5 5 4 10 10 10 10 4 4 4
2 P 1A 1A 1A 2A 2A 2A 5A 4B1 4B-1 4B-1 4B1 5A 10A 10A
5 P 1A 2A 2B 4A 4B1 4B-1 1A 8B-1 8B1 8A-1 8A1 2A 4A 4A
Type
80.29.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.29.1b R 1 1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1
80.29.1c R 1 1 −1 −1 1 1 1 1 1 −1 −1 1 −1 −1
80.29.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1
80.29.1e1 C 1 1 −1 1 −1 −1 1 −i i i −i 1 1 1
80.29.1e2 C 1 1 −1 1 −1 −1 1 i −i −i i 1 1 1
80.29.1f1 C 1 1 1 −1 −1 −1 1 −i i −i i 1 −1 −1
80.29.1f2 C 1 1 1 −1 −1 −1 1 i −i i −i 1 −1 −1
80.29.2a1 C 2 −2 0 0 −2i 2i 2 0 0 0 0 −2 0 0
80.29.2a2 C 2 −2 0 0 2i −2i 2 0 0 0 0 −2 0 0
80.29.4a R 4 4 0 4 0 0 −1 0 0 0 0 −1 −1 −1
80.29.4b R 4 4 0 −4 0 0 −1 0 0 0 0 −1 1 1
80.29.4c1 C 4 −4 0 0 0 0 −1 0 0 0 0 1 −2ζ203+ζ205−2ζ207 2ζ203−ζ205+2ζ207
80.29.4c2 C 4 −4 0 0 0 0 −1 0 0 0 0 1 2ζ203−ζ205+2ζ207 −2ζ203+ζ205−2ζ207

magma: CharacterTable(G);