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Properties

Label 40T8
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(40, 8);
 

Group action invariants

Degree $n$:  $40$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:Q_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $40$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,38,35,31,28,24,19,16,12,7,2,37,36,32,27,23,20,15,11,8)(3,40,33,30,26,21,18,14,10,5,4,39,34,29,25,22,17,13,9,6), (1,4,2,3)(5,37,6,38)(7,40,8,39)(9,35,10,36)(11,34,12,33)(13,31,14,32)(15,29,16,30)(17,28,18,27)(19,26,20,25)(21,23,22,24)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $Q_8$
$10$:  $D_{5}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $D_{5}$

Degree 8: $Q_8$

Degree 10: $D_5$, $D_{10}$ x 2

Degree 20: 20T4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  40.4
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A 4B 4C 5A1 5A2 10A1 10A3 20A1 20A3 20A7 20A9
Size 1 1 2 10 10 2 2 2 2 2 2 2 2
2 P 1A 1A 2A 2A 2A 5A2 5A1 5A1 5A2 10A1 10A3 10A3 10A1
5 P 1A 2A 4A 4B 4C 1A 1A 2A 2A 4A 4A 4A 4A
Type
40.4.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
40.4.1b R 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
40.4.1c R 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
40.4.1d R 1 1 1 −1 −1 1 1 1 1 1 1 1 1
40.4.2a S 2 −2 0 0 0 2 2 −2 −2 0 0 0 0
40.4.2b1 R 2 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
40.4.2b2 R 2 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
40.4.2c1 R 2 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
40.4.2c2 R 2 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
40.4.2d1 S 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 −ζ20−3−ζ203 ζ20−1+ζ20 −ζ20−1−ζ20 ζ20−3+ζ203
40.4.2d2 S 2 −2 0 0 0 −ζ20−2−ζ202 ζ20−4+ζ204 −ζ20−4−ζ204 ζ20−2+ζ202 ζ20−3+ζ203 −ζ20−1−ζ20 ζ20−1+ζ20 −ζ20−3−ζ203
40.4.2d3 S 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 −ζ20−1−ζ20 −ζ20−3−ζ203 ζ20−3+ζ203 ζ20−1+ζ20
40.4.2d4 S 2 −2 0 0 0 ζ20−4+ζ204 −ζ20−2−ζ202 ζ20−2+ζ202 −ζ20−4−ζ204 ζ20−1+ζ20 ζ20−3+ζ203 −ζ20−3−ζ203 −ζ20−1−ζ20

magma: CharacterTable(G);