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Properties

Label 8T17
8T17 1 2 1->2 5 1->5 3 2->3 6 2->6 7 3->7 8 3->8 4 4->8 8->1
Degree $8$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\wr C_2$

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Copy content magma:G := TransitiveGroup(8, 17);
 

Group invariants

Abstract group:  $C_4\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $32=2^{5}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $17$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[4^{2}]2$
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:  $(1,2,3,8)$, $(1,5)(2,6)(3,7)(4,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T17, 16T28, 16T42, 32T14

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 $2^{4}$ $1$ $2$ $4$ $(1,3)(2,8)(4,6)(5,7)$
2B $2^{2},1^{4}$ $2$ $2$ $2$ $(1,3)(2,8)$
2C $2^{4}$ $4$ $2$ $4$ $(1,7)(2,4)(3,5)(6,8)$
4A1 $4^{2}$ $1$ $4$ $6$ $(1,2,3,8)(4,5,6,7)$
4A-1 $4^{2}$ $1$ $4$ $6$ $(1,8,3,2)(4,7,6,5)$
4B $4^{2}$ $2$ $4$ $6$ $(1,8,3,2)(4,5,6,7)$
4C1 $4,1^{4}$ $2$ $4$ $3$ $(1,2,3,8)$
4C-1 $4,1^{4}$ $2$ $4$ $3$ $(1,8,3,2)$
4D1 $4,2^{2}$ $2$ $4$ $5$ $(1,3)(2,8)(4,5,6,7)$
4D-1 $4,2^{2}$ $2$ $4$ $5$ $(1,3)(2,8)(4,7,6,5)$
4E $4^{2}$ $4$ $4$ $6$ $(1,5,3,7)(2,6,8,4)$
8A1 $8$ $4$ $8$ $7$ $(1,4,2,5,3,6,8,7)$
8A-1 $8$ $4$ $8$ $7$ $(1,6,8,5,3,4,2,7)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B 4C1 4C-1 4D1 4D-1 4E 8A1 8A-1
Size 1 1 2 4 1 1 2 2 2 2 2 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2A 2B 2B 2B 2B 2A 4A1 4A-1
Type
32.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.11.1b R 1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 1 1
32.11.1c R 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1
32.11.1d R 1 1 1 1 1 1 −1 −1 −1 1 −1 1 −1 −1
32.11.1e1 C 1 1 −1 −1 −1 −1 −i −i i 1 i 1 i −i
32.11.1e2 C 1 1 −1 −1 −1 −1 i i −i 1 −i 1 −i i
32.11.1f1 C 1 1 −1 1 −1 −1 −i −i i 1 i −1 −i i
32.11.1f2 C 1 1 −1 1 −1 −1 i i −i 1 −i −1 i −i
32.11.2a R 2 2 −2 0 2 2 0 0 0 −2 0 0 0 0
32.11.2b R 2 2 2 0 −2 −2 0 0 0 −2 0 0 0 0
32.11.2c1 C 2 −2 0 0 −2i 2i −1+i 1−i 1+i 0 −1−i 0 0 0
32.11.2c2 C 2 −2 0 0 2i −2i −1−i 1+i 1−i 0 −1+i 0 0 0
32.11.2d1 C 2 −2 0 0 −2i 2i 1−i −1+i −1−i 0 1+i 0 0 0
32.11.2d2 C 2 −2 0 0 2i −2i 1+i −1−i −1+i 0 1−i 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $t x^{8} + x^{7} - 7 x^{5} + 14 t x^{4} + 7 x^{3} - x + t$ Copy content Toggle raw display