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Properties

Label 40T32
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, 32);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:D_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,4)(5,40)(6,39)(7,37)(8,38)(9,33)(10,34)(11,36)(12,35)(13,32)(14,31)(15,29)(16,30)(17,27)(18,28)(19,26)(20,25)(21,24)(22,23), (1,38,4,39,2,37,3,40)(5,34,7,35,6,33,8,36)(9,32,12,29,10,31,11,30)(13,27,16,26,14,28,15,25)(17,23,20,22,18,24,19,21)
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: $D_{10}$

Degree 20: 20T7

Low degree siblings

40T48

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

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:  80.15
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A2 8A1 8A3 10A1 10A3 10B1 10B-1 10B3 10B-3 20A1 20A3
Size 1 1 4 20 2 2 2 10 10 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 2A 5A2 5A1 4A 4A 5A2 5A1 5A2 5A2 5A1 5A1 10A1 10A3
5 P 1A 2A 2B 2C 4A 1A 1A 8A3 8A1 2A 2A 2B 2B 2B 2B 4A 4A
Type
80.15.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.15.1b R 1 1 −1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1
80.15.1c R 1 1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
80.15.1d R 1 1 1 −1 1 1 1 −1 −1 1 1 1 1 1 1 1 1
80.15.2a R 2 2 0 0 −2 2 2 0 0 2 2 0 0 0 0 −2 −2
80.15.2b1 R 2 2 2 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
80.15.2b2 R 2 2 2 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
80.15.2c1 R 2 −2 0 0 0 2 2 −ζ8−1−ζ8 ζ8−1+ζ8 −2 −2 0 0 0 0 0 0
80.15.2c2 R 2 −2 0 0 0 2 2 ζ8−1+ζ8 −ζ8−1−ζ8 −2 −2 0 0 0 0 0 0
80.15.2d1 R 2 2 −2 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52
80.15.2d2 R 2 2 −2 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5
80.15.2e1 C 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
80.15.2e2 C 2 2 0 0 −2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−1−ζ5 −ζ5−2−ζ52
80.15.2e3 C 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2−ζ52 −ζ5−2+ζ52 −ζ5−2−1−2ζ5−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.15.2e4 C 2 2 0 0 −2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2+ζ52 ζ5−2−ζ52 ζ5−2+1+2ζ5+ζ52 −ζ5−2−1−2ζ5−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
80.15.4a1 R 4 −4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 0 0 0 0 0 0
80.15.4a2 R 4 −4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 0 0 0 0 0 0

magma: CharacterTable(G);