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Properties

Label 40T321
Degree $40$
Order $400$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_5^2:Q_8$

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Copy content magma:G := TransitiveGroup(40, 321);
 

Group invariants

Abstract group:  $C_2\times C_5^2:Q_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $400=2^{4} \cdot 5^{2}$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $321$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,26,34,12)(2,25,33,11)(3,28,35,10)(4,27,36,9)(5,7,6,8)(13,30,38,21)(14,29,37,22)(15,32,39,24)(16,31,40,23)(17,20,18,19)$, $(1,24,34,13)(2,23,33,14)(3,22,35,15)(4,21,36,16)(5,9,32,27)(6,10,31,28)(7,11,30,25)(8,12,29,26)(17,40,18,39)(19,37,20,38)$, $(1,16,9,40)(2,15,10,39)(3,13,12,38)(4,14,11,37)(5,26,6,25)(7,27,8,28)(17,24,36,32)(18,23,35,31)(19,21,33,30)(20,22,34,29)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$, $Q_8$ x 2
$16$:  $Q_8\times C_2$
$200$:  $C_5^2 : Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 8: $Q_8$

Degree 10: $C_5^2 : Q_8$

Degree 20: $C_5^2:Q_8$

Low degree siblings

20T99 x 6, 40T321 x 2, 40T325 x 3

Siblings are shown 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 $2^{20}$ $1$ $2$ $20$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)$
2B $2^{20}$ $25$ $2$ $20$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5, 6)( 7, 8)( 9,10)(11,12)(13,38)(14,37)(15,39)(16,40)(21,30)(22,29)(23,31)(24,32)(25,36)(26,35)(27,34)(28,33)$
2C $2^{16},1^{8}$ $25$ $2$ $16$ $( 1,27)( 2,28)( 3,26)( 4,25)( 5,14)( 6,13)( 7,15)( 8,16)( 9,19)(10,20)(11,18)(12,17)(21,40)(22,39)(23,37)(24,38)$
4A $4^{10}$ $50$ $4$ $30$ $( 1,14,19,37)( 2,13,20,38)( 3,16,17,40)( 4,15,18,39)( 5, 9, 6,10)( 7,11, 8,12)(21,26,30,35)(22,25,29,36)(23,27,31,34)(24,28,32,33)$
4B $4^{10}$ $50$ $4$ $30$ $( 1,29, 2,30)( 3,31, 4,32)( 5,35,13,11)( 6,36,14,12)( 7,34,16,10)( 8,33,15, 9)(17,23,26,38)(18,24,25,37)(19,22,27,40)(20,21,28,39)$
4C $4^{10}$ $50$ $4$ $30$ $( 1,11,34,25)( 2,12,33,26)( 3, 9,35,27)( 4,10,36,28)( 5, 7, 6, 8)(13,22,38,29)(14,21,37,30)(15,23,39,31)(16,24,40,32)(17,20,18,19)$
4D $4^{10}$ $50$ $4$ $30$ $( 1,35, 9,18)( 2,36,10,17)( 3,33,12,19)( 4,34,11,20)( 5,16,32,21)( 6,15,31,22)( 7,14,30,23)( 8,13,29,24)(25,28,26,27)(37,40,38,39)$
4E $4^{10}$ $50$ $4$ $30$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,33,38)(10,14,34,37)(11,16,35,40)(12,15,36,39)(17,21,26,30)(18,22,25,29)(19,23,27,31)(20,24,28,32)$
4F $4^{10}$ $50$ $4$ $30$ $( 1, 7,28,39)( 2, 8,27,40)( 3, 5,25,37)( 4, 6,26,38)( 9,30,20,15)(10,29,19,16)(11,31,17,13)(12,32,18,14)(21,34,22,33)(23,36,24,35)$
5A $5^{4},1^{20}$ $8$ $5$ $16$ $( 5,24,38,14,31)( 6,23,37,13,32)( 7,21,40,15,29)( 8,22,39,16,30)$
5B $5^{8}$ $8$ $5$ $32$ $( 1,33,27,20,10)( 2,34,28,19, 9)( 3,36,26,18,11)( 4,35,25,17,12)( 5,14,24,31,38)( 6,13,23,32,37)( 7,15,21,29,40)( 8,16,22,30,39)$
5C $5^{8}$ $8$ $5$ $32$ $( 1,33,27,20,10)( 2,34,28,19, 9)( 3,36,26,18,11)( 4,35,25,17,12)( 5,31,14,38,24)( 6,32,13,37,23)( 7,29,15,40,21)( 8,30,16,39,22)$
10A $10^{2},2^{10}$ $8$ $10$ $28$ $( 1, 2)( 3, 4)( 5,13,24,32,38, 6,14,23,31,37)( 7,16,21,30,40, 8,15,22,29,39)( 9,10)(11,12)(17,18)(19,20)(25,26)(27,28)(33,34)(35,36)$
10B $10^{4}$ $8$ $10$ $36$ $( 1,19,33, 9,27, 2,20,34,10,28)( 3,17,36,12,26, 4,18,35,11,25)( 5,32,14,37,24, 6,31,13,38,23)( 7,30,15,39,21, 8,29,16,40,22)$
10C $10^{4}$ $8$ $10$ $36$ $( 1,19,33, 9,27, 2,20,34,10,28)( 3,17,36,12,26, 4,18,35,11,25)( 5,37,31,23,14, 6,38,32,24,13)( 7,39,29,22,15, 8,40,30,21,16)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 10A 10B 10C
Size 1 1 25 25 50 50 50 50 50 50 8 8 8 8 8 8
2 P 1A 1A 1A 1A 2B 2B 2B 2B 2B 2B 5A 5B 5C 5A 5B 5C
5 P 1A 2A 2B 2C 4A 4B 4C 4D 4E 4F 1A 1A 1A 2A 2A 2A
Type
400.212.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
400.212.1b R 1 −1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1
400.212.1c R 1 −1 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 −1 −1
400.212.1d R 1 −1 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1
400.212.1e R 1 −1 1 −1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1
400.212.1f R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1
400.212.1g R 1 1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1
400.212.1h R 1 1 1 1 1 −1 −1 −1 1 −1 1 1 1 1 1 1
400.212.2a S 2 −2 −2 2 0 0 0 0 0 0 2 2 2 −2 −2 −2
400.212.2b S 2 2 −2 −2 0 0 0 0 0 0 2 2 2 2 2 2
400.212.8a R 8 8 0 0 0 0 0 0 0 0 −2 −2 3 −2 −2 3
400.212.8b R 8 8 0 0 0 0 0 0 0 0 −2 3 −2 −2 3 −2
400.212.8c R 8 8 0 0 0 0 0 0 0 0 3 −2 −2 3 −2 −2
400.212.8d R 8 −8 0 0 0 0 0 0 0 0 −2 −2 3 2 2 −3
400.212.8e R 8 −8 0 0 0 0 0 0 0 0 −2 3 −2 2 −3 2
400.212.8f R 8 −8 0 0 0 0 0 0 0 0 3 −2 −2 −3 2 2

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed