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Properties

Label 8T31
8T31 1 3 1->3 8 1->8 2 2->3 2->8 4 5 4->5 6 4->6 4->8 7 5->7 6->7
Degree $8$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(((C_4 \times C_2): C_2):C_2):C_2$

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Copy content magma:G := TransitiveGroup(8, 31);
 

Group invariants

Abstract group:  $(((C_4 \times C_2): C_2):C_2):C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{4}]E(4)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3)(2,8)(4,6)(5,7)$, $(1,8)(2,3)(4,5)(6,7)$, $(4,8)$
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
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Low degree siblings

8T29 x 6, 8T31, 16T127, 16T128 x 3, 16T129 x 3, 16T147, 16T149 x 6, 16T150 x 3, 32T136 x 3, 32T137 x 2, 32T163 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 $1^{8}$ $1$ $1$ $0$ $()$
2A $2^{4}$ $1$ $2$ $4$ $(1,5)(2,6)(3,7)(4,8)$
2B $2^{2},1^{4}$ $2$ $2$ $2$ $(2,6)(3,7)$
2C $2^{2},1^{4}$ $2$ $2$ $2$ $(1,5)(3,7)$
2D $2^{2},1^{4}$ $2$ $2$ $2$ $(3,7)(4,8)$
2E $2,1^{6}$ $4$ $2$ $1$ $(3,7)$
2F $2^{3},1^{2}$ $4$ $2$ $3$ $(1,5)(2,6)(3,7)$
2G $2^{4}$ $4$ $2$ $4$ $(1,8)(2,3)(4,5)(6,7)$
2H $2^{4}$ $4$ $2$ $4$ $(1,3)(2,8)(4,6)(5,7)$
2I $2^{4}$ $4$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
4A $4^{2}$ $4$ $4$ $6$ $(1,4,5,8)(2,3,6,7)$
4B $4^{2}$ $4$ $4$ $6$ $(1,3,5,7)(2,4,6,8)$
4C $4^{2}$ $4$ $4$ $6$ $(1,2,5,6)(3,4,7,8)$
4D $4,2^{2}$ $8$ $4$ $5$ $(1,8)(2,3,6,7)(4,5)$
4E $4,2^{2}$ $8$ $4$ $5$ $(1,3,5,7)(2,8)(4,6)$
4F $4,2^{2}$ $8$ $4$ $5$ $(1,2)(3,4,7,8)(5,6)$

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F
Size 1 1 2 2 2 4 4 4 4 4 4 4 4 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2B 2C 2D
Type
64.138.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.138.1b R 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 1 1
64.138.1c R 1 1 1 1 1 −1 −1 −1 1 −1 −1 1 −1 1 −1 1
64.138.1d R 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 1 −1
64.138.1e R 1 1 1 1 1 −1 1 −1 1 1 1 1 1 −1 −1 −1
64.138.1f R 1 1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1
64.138.1g R 1 1 1 1 1 1 −1 1 1 −1 −1 1 −1 −1 1 −1
64.138.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1
64.138.2a R 2 2 −2 −2 2 0 −2 0 0 0 0 0 2 0 0 0
64.138.2b R 2 2 −2 −2 2 0 2 0 0 0 0 0 −2 0 0 0
64.138.2c R 2 2 −2 2 −2 0 0 0 −2 0 0 2 0 0 0 0
64.138.2d R 2 2 −2 2 −2 0 0 0 2 0 0 −2 0 0 0 0
64.138.2e R 2 2 2 −2 −2 0 0 0 0 −2 2 0 0 0 0 0
64.138.2f R 2 2 2 −2 −2 0 0 0 0 2 −2 0 0 0 0 0
64.138.4a R 4 −4 0 0 0 −2 0 2 0 0 0 0 0 0 0 0
64.138.4b R 4 −4 0 0 0 2 0 −2 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + 4 x^{6} + \left(t + 4\right) x^{4} + 2 t x^{2} + \left(t + 8\right)$ Copy content Toggle raw display