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Properties

Label 40T5
Degree $40$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times Q_8$

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Copy content magma:G := TransitiveGroup(40, 5);
 

Group invariants

Abstract group:  $C_5\times Q_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $40=2^{3} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$8$:  $Q_8$
$10$:  $C_{10}$ x 3
$20$:  20T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: $C_5$

Degree 8: $Q_8$

Degree 10: $C_{10}$ x 3

Degree 20: 20T3

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 TypeSizeOrderIndexRepresentative
1A $1^{40}$ $1$ $1$ $0$ $()$
2A $2^{20}$ $1$ $2$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
4A $4^{10}$ $2$ $4$ $30$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,20,18,19)(21,24,22,23)(25,27,26,28)(29,32,30,31)(33,35,34,36)(37,40,38,39)$
4B $4^{10}$ $2$ $4$ $30$ $( 1,24, 2,23)( 3,21, 4,22)( 5,25, 6,26)( 7,28, 8,27)( 9,30,10,29)(11,31,12,32)(13,35,14,36)(15,33,16,34)(17,39,18,40)(19,37,20,38)$
4C $4^{10}$ $2$ $4$ $30$ $( 1,22, 2,21)( 3,24, 4,23)( 5,27, 6,28)( 7,25, 8,26)( 9,31,10,32)(11,29,12,30)(13,34,14,33)(15,35,16,36)(17,37,18,38)(19,40,20,39)$
5A1 $5^{8}$ $1$ $5$ $32$ $( 1,18,35,10,28)( 2,17,36, 9,27)( 3,19,34,11,25)( 4,20,33,12,26)( 5,22,38,16,32)( 6,21,37,15,31)( 7,23,39,13,30)( 8,24,40,14,29)$
5A-1 $5^{8}$ $1$ $5$ $32$ $( 1,28,10,35,18)( 2,27, 9,36,17)( 3,25,11,34,19)( 4,26,12,33,20)( 5,32,16,38,22)( 6,31,15,37,21)( 7,30,13,39,23)( 8,29,14,40,24)$
5A2 $5^{8}$ $1$ $5$ $32$ $( 1,35,28,18,10)( 2,36,27,17, 9)( 3,34,25,19,11)( 4,33,26,20,12)( 5,38,32,22,16)( 6,37,31,21,15)( 7,39,30,23,13)( 8,40,29,24,14)$
5A-2 $5^{8}$ $1$ $5$ $32$ $( 1,10,18,28,35)( 2, 9,17,27,36)( 3,11,19,25,34)( 4,12,20,26,33)( 5,16,22,32,38)( 6,15,21,31,37)( 7,13,23,30,39)( 8,14,24,29,40)$
10A1 $10^{4}$ $1$ $10$ $36$ $( 1, 9,18,27,35, 2,10,17,28,36)( 3,12,19,26,34, 4,11,20,25,33)( 5,15,22,31,38, 6,16,21,32,37)( 7,14,23,29,39, 8,13,24,30,40)$
10A-1 $10^{4}$ $1$ $10$ $36$ $( 1,36,28,17,10, 2,35,27,18, 9)( 3,33,25,20,11, 4,34,26,19,12)( 5,37,32,21,16, 6,38,31,22,15)( 7,40,30,24,13, 8,39,29,23,14)$
10A3 $10^{4}$ $1$ $10$ $36$ $( 1,27,10,36,18, 2,28, 9,35,17)( 3,26,11,33,19, 4,25,12,34,20)( 5,31,16,37,22, 6,32,15,38,21)( 7,29,13,40,23, 8,30,14,39,24)$
10A-3 $10^{4}$ $1$ $10$ $36$ $( 1,17,35, 9,28, 2,18,36,10,27)( 3,20,34,12,25, 4,19,33,11,26)( 5,21,38,15,32, 6,22,37,16,31)( 7,24,39,14,30, 8,23,40,13,29)$
20A1 $20^{2}$ $2$ $20$ $38$ $( 1,25, 9,33,18, 3,27,12,35,19, 2,26,10,34,17, 4,28,11,36,20)( 5,30,15,40,22, 7,31,14,38,23, 6,29,16,39,21, 8,32,13,37,24)$
20A-1 $20^{2}$ $2$ $20$ $38$ $( 1,19,36,12,28, 3,17,33,10,25, 2,20,35,11,27, 4,18,34, 9,26)( 5,23,37,14,32, 7,21,40,16,30, 6,24,38,13,31, 8,22,39,15,29)$
20A3 $20^{2}$ $2$ $20$ $38$ $( 1,33,27,19,10, 4,36,25,18,12, 2,34,28,20, 9, 3,35,26,17,11)( 5,40,31,23,16, 8,37,30,22,14, 6,39,32,24,15, 7,38,29,21,13)$
20A-3 $20^{2}$ $2$ $20$ $38$ $( 1,11,17,26,35, 3, 9,20,28,34, 2,12,18,25,36, 4,10,19,27,33)( 5,13,21,29,38, 7,15,24,32,39, 6,14,22,30,37, 8,16,23,31,40)$
20B1 $20^{2}$ $2$ $20$ $38$ $( 1,40,36,30,28,24,17,13,10, 8, 2,39,35,29,27,23,18,14, 9, 7)( 3,37,33,32,25,21,20,16,11, 6, 4,38,34,31,26,22,19,15,12, 5)$
20B-1 $20^{2}$ $2$ $20$ $38$ $( 1, 8, 9,13,18,24,27,30,35,40, 2, 7,10,14,17,23,28,29,36,39)( 3, 6,12,16,19,21,26,32,34,37, 4, 5,11,15,20,22,25,31,33,38)$
20B3 $20^{2}$ $2$ $20$ $38$ $( 1,30,17, 8,35,23, 9,40,28,13, 2,29,18, 7,36,24,10,39,27,14)( 3,32,20, 6,34,22,12,37,25,16, 4,31,19, 5,33,21,11,38,26,15)$
20B-3 $20^{2}$ $2$ $20$ $38$ $( 1,14,27,39,10,24,36, 7,18,29, 2,13,28,40, 9,23,35, 8,17,30)( 3,15,26,38,11,21,33, 5,19,31, 4,16,25,37,12,22,34, 6,20,32)$
20C1 $20^{2}$ $2$ $20$ $38$ $( 1, 5, 9,15,18,22,27,31,35,38, 2, 6,10,16,17,21,28,32,36,37)( 3, 8,12,13,19,24,26,30,34,40, 4, 7,11,14,20,23,25,29,33,39)$
20C-1 $20^{2}$ $2$ $20$ $38$ $( 1,38,36,31,28,22,17,15,10, 5, 2,37,35,32,27,21,18,16, 9, 6)( 3,40,33,30,25,24,20,13,11, 8, 4,39,34,29,26,23,19,14,12, 7)$
20C3 $20^{2}$ $2$ $20$ $38$ $( 1,15,27,38,10,21,36, 5,18,31, 2,16,28,37, 9,22,35, 6,17,32)( 3,13,26,40,11,23,33, 8,19,30, 4,14,25,39,12,24,34, 7,20,29)$
20C-3 $20^{2}$ $2$ $20$ $38$ $( 1,32,17, 6,35,22, 9,37,28,16, 2,31,18, 5,36,21,10,38,27,15)( 3,29,20, 7,34,24,12,39,25,14, 4,30,19, 8,33,23,11,40,26,13)$

Malle's constant $a(G)$:     $1/20$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 4A 4B 4C 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20B1 20B-1 20B3 20B-3 20C1 20C-1 20C3 20C-3
Size 1 1 2 2 2 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 2A 2A 2A 5A2 5A-2 5A-1 5A1 5A1 5A-1 5A-2 5A2 10A1 10A-1 10A3 10A-3 10A-1 10A1 10A-3 10A3 10A1 10A-1 10A3 10A-3
5 P 1A 2A 4A 4B 4C 1A 1A 1A 1A 2A 2A 2A 2A 4A 4A 4A 4A 4B 4B 4B 4B 4C 4C 4C 4C
Type
40.11.1a R 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
40.11.1b R 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
40.11.1c R 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
40.11.1d R 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
40.11.1e1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−2 ζ5−1 ζ5 ζ52 ζ52 ζ5−1 ζ5−2 ζ5−1 ζ5−2 ζ5 ζ52
40.11.1e2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ52 ζ5 ζ5−1 ζ5−2 ζ5−2 ζ5 ζ52 ζ5 ζ52 ζ5−1 ζ5−2
40.11.1e3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ5−1 ζ52 ζ5−2 ζ5 ζ5 ζ52 ζ5−1 ζ52 ζ5−1 ζ5−2 ζ5
40.11.1e4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5 ζ5−2 ζ52 ζ5−1 ζ5−1 ζ5−2 ζ5 ζ5−2 ζ5 ζ52 ζ5−1
40.11.1f1 C 1 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−2 ζ5−1 −ζ5 ζ52 −ζ52 −ζ5−1 −ζ5−2 −ζ5−1 ζ5−2 ζ5 −ζ52
40.11.1f2 C 1 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ52 ζ5 −ζ5−1 ζ5−2 −ζ5−2 −ζ5 −ζ52 −ζ5 ζ52 ζ5−1 −ζ5−2
40.11.1f3 C 1 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ5−1 ζ52 −ζ5−2 ζ5 −ζ5 −ζ52 −ζ5−1 −ζ52 ζ5−1 ζ5−2 −ζ5
40.11.1f4 C 1 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5 ζ5−2 −ζ52 ζ5−1 −ζ5−1 −ζ5−2 −ζ5 −ζ5−2 ζ5 ζ52 −ζ5−1
40.11.1g1 C 1 1 −1 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 ζ5−2 −ζ5−1 ζ5 −ζ52 ζ52 ζ5−1 −ζ5−2 −ζ5−1 −ζ5−2 −ζ5 −ζ52
40.11.1g2 C 1 1 −1 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 ζ52 −ζ5 ζ5−1 −ζ5−2 ζ5−2 ζ5 −ζ52 −ζ5 −ζ52 −ζ5−1 −ζ5−2
40.11.1g3 C 1 1 −1 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 ζ5−1 −ζ52 ζ5−2 −ζ5 ζ5 ζ52 −ζ5−1 −ζ52 −ζ5−1 −ζ5−2 −ζ5
40.11.1g4 C 1 1 −1 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 ζ5 −ζ5−2 ζ52 −ζ5−1 ζ5−1 ζ5−2 −ζ5 −ζ5−2 −ζ5 −ζ52 −ζ5−1
40.11.1h1 C 1 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 −ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ52 −ζ5−1 ζ5−2 ζ5−1 −ζ5−2 −ζ5 ζ52
40.11.1h2 C 1 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 −ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ5−2 −ζ5 ζ52 ζ5 −ζ52 −ζ5−1 ζ5−2
40.11.1h3 C 1 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 −ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5 −ζ52 ζ5−1 ζ52 −ζ5−1 −ζ5−2 ζ5
40.11.1h4 C 1 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 −ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5−1 −ζ5−2 ζ5 ζ5−2 −ζ5 −ζ52 ζ5−1
40.11.2a S 2 −2 0 0 0 2 2 2 2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 0 0 0 0
40.11.2b1 C 2 −2 0 0 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 −2ζ52 −2ζ5−2 −2ζ5−1 −2ζ5 0 0 0 0 0 0 0 0 0 0 0 0
40.11.2b2 C 2 −2 0 0 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 −2ζ5−2 −2ζ52 −2ζ5 −2ζ5−1 0 0 0 0 0 0 0 0 0 0 0 0
40.11.2b3 C 2 −2 0 0 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 −2ζ5 −2ζ5−1 −2ζ52 −2ζ5−2 0 0 0 0 0 0 0 0 0 0 0 0
40.11.2b4 C 2 −2 0 0 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 −2ζ5−1 −2ζ5 −2ζ5−2 −2ζ52 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed