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Properties

Label 40T325
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, 325);
 

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 $t$:  $325$
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:  $(1,40,9,22)(2,39,10,21)(3,38,12,24)(4,37,11,23)(5,36,13,17)(6,35,14,18)(7,34,15,20)(8,33,16,19)(25,31)(26,32)(27,30)(28,29)$, $(1,32,19,14)(2,31,20,13)(3,29,18,16)(4,30,17,15)(5,34,37,27)(6,33,38,28)(7,36,39,25)(8,35,40,26)(9,24)(10,23)(11,21)(12,22)$, $(1,25,19,36)(2,26,20,35)(3,27,18,34)(4,28,17,33)(5,8)(6,7)(9,11)(10,12)(13,22,37,29)(14,21,38,30)(15,24,39,32)(16,23,40,31)$
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 7

Degree 4: $C_2^2$ x 7

Degree 5: None

Degree 8: $C_2^3$

Degree 10: $C_5^2 : Q_8$

Degree 20: 20T47, 20T99 x 2

Low degree siblings

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

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

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