↑

Properties

Label 8T26
8T26 1 2 1->2 5 1->5 7 1->7 3 2->3 4 3->4 3->5 4->5 8 4->8 4->8 6 5->6 6->7 7->8 8->1
Degree $8$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_4^2 : C_2):C_2$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(8, 26);
 

Group invariants

Abstract group:  $(C_4^2 : C_2):C_2$
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 $t$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[2^{4}]eD(4)$
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:  $(1,2,3,4,5,6,7,8)$, $(1,7)(3,5)(4,8)$, $(1,5)(4,8)$
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: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T26 x 3, 16T135 x 2, 16T141 x 2, 16T142 x 2, 16T152 x 2, 32T147 x 2, 32T148 x 2, 32T155, 32T156

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 $2^{2},1^{4}$ $2$ $2$ $2$ $(1,5)(3,7)$
2C $2^{2},1^{4}$ $4$ $2$ $2$ $(3,7)(4,8)$
2D $2^{4}$ $4$ $2$ $4$ $(1,7)(2,8)(3,5)(4,6)$
2E $2^{4}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2F $2^{4}$ $4$ $2$ $4$ $(1,2)(3,8)(4,7)(5,6)$
2G $2^{3},1^{2}$ $8$ $2$ $3$ $(1,7)(3,5)(4,8)$
4A $4^{2}$ $2$ $4$ $6$ $(1,7,5,3)(2,8,6,4)$
4B $4^{2}$ $2$ $4$ $6$ $(1,3,5,7)(2,8,6,4)$
4C $4,1^{4}$ $4$ $4$ $3$ $(1,7,5,3)$
4D $4,2^{2}$ $4$ $4$ $5$ $(1,7,5,3)(2,6)(4,8)$
4E $4^{2}$ $4$ $4$ $6$ $(1,6,5,2)(3,8,7,4)$
4F $4^{2}$ $4$ $4$ $6$ $(1,6,5,2)(3,4,7,8)$
8A $8$ $8$ $8$ $7$ $(1,8,7,6,5,4,3,2)$
8B $8$ $8$ $8$ $7$ $(1,8,3,6,5,4,7,2)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 8A 8B
Size 1 1 2 4 4 4 4 8 2 2 4 4 4 4 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2B 2A 2A 4A 4B
Type
64.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.134.1b R 1 1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1 1 −1
64.134.1c R 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 1 −1 −1 1
64.134.1d R 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
64.134.1e R 1 1 1 −1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1
64.134.1f R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 −1 1 −1 1
64.134.1g R 1 1 1 1 −1 −1 −1 1 1 1 −1 −1 −1 1 1 −1
64.134.1h R 1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1
64.134.2a R 2 2 −2 −2 0 0 0 0 2 −2 0 0 0 2 0 0
64.134.2b R 2 2 −2 0 0 0 −2 0 −2 2 0 0 2 0 0 0
64.134.2c R 2 2 −2 0 0 0 2 0 −2 2 0 0 −2 0 0 0
64.134.2d R 2 2 −2 2 0 0 0 0 2 −2 0 0 0 −2 0 0
64.134.2e R 2 2 2 0 −2 2 0 0 −2 −2 0 0 0 0 0 0
64.134.2f R 2 2 2 0 2 −2 0 0 −2 −2 0 0 0 0 0 0
64.134.4a R 4 −4 0 0 0 0 0 0 0 0 −2 2 0 0 0 0
64.134.4b R 4 −4 0 0 0 0 0 0 0 0 2 −2 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} + 4 x^{6} + \left(3 t + 5\right) x^{4} + \left(6 t + 2\right) x^{2} - t$ Copy content Toggle raw display