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Properties

Label 40T25
Degree $40$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:\OD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(40, 25);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:\OD_{16}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $20$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,30,33,8,2,29,34,7)(3,32,36,5,4,31,35,6)(9,13,28,23,10,14,27,24)(11,16,26,21,12,15,25,22)(17,40,19,37,18,39,20,38), (1,9,2,10)(3,11,4,12)(5,7,6,8)(13,37,14,38)(15,40,16,39)(17,36,18,35)(19,34,20,33)(21,29,22,30)(23,32,24,31)(25,27,26,28)
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: 20T5

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

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:  80.33
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A1 4A-1 4B 5A 8A1 8A-1 8B1 8B-1 10A 10B1 10B3
Size 1 1 2 5 5 10 4 10 10 10 10 4 4 4
2 P 1A 1A 1A 2A 2A 2A 5A 4A1 4A-1 4A-1 4A1 5A 5A 5A
5 P 1A 2A 2B 4A1 4A-1 4B 1A 8B-1 8B1 8A-1 8A1 2A 2B 2B
Type
80.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.33.1b R 1 1 −1 1 1 −1 1 −1 −1 1 1 1 −1 −1
80.33.1c R 1 1 −1 1 1 −1 1 1 1 −1 −1 1 −1 −1
80.33.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1
80.33.1e1 C 1 1 −1 −1 −1 1 1 −i i i −i 1 −1 −1
80.33.1e2 C 1 1 −1 −1 −1 1 1 i −i −i i 1 −1 −1
80.33.1f1 C 1 1 1 −1 −1 −1 1 −i i −i i 1 1 1
80.33.1f2 C 1 1 1 −1 −1 −1 1 i −i i −i 1 1 1
80.33.2a1 C 2 −2 0 −2i 2i 0 2 0 0 0 0 −2 0 0
80.33.2a2 C 2 −2 0 2i −2i 0 2 0 0 0 0 −2 0 0
80.33.4a R 4 4 4 0 0 0 −1 0 0 0 0 −1 −1 −1
80.33.4b R 4 4 −4 0 0 0 −1 0 0 0 0 −1 1 1
80.33.4c1 S 4 −4 0 0 0 0 −1 0 0 0 0 1 2ζ5−2+1+2ζ52 −2ζ5−2−1−2ζ52
80.33.4c2 S 4 −4 0 0 0 0 −1 0 0 0 0 1 −2ζ5−2−1−2ζ52 2ζ5−2+1+2ζ52

magma: CharacterTable(G);