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Properties

Label 8T27
8T27 1 2 1->2 3 2->3 8 3->8 4 5 4->5 4->8 6 5->6 7 6->7 7->4 8->1
Degree $8$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $((C_8 : C_2):C_2):C_2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(8, 27);
 

Group invariants

Abstract group:  $((C_8 : C_2):C_2):C_2$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $8$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[2^{4}]4$
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:  $(1,2,3,8)(4,5,6,7)$, $(4,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Low degree siblings

8T27, 8T28 x 2, 16T130, 16T157 x 2, 16T158 x 2, 16T159 x 2, 16T166, 16T170, 16T171, 16T172, 32T138 x 2, 32T139, 32T170, 32T176

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

Malle's constant $a(G)$:     $1$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 4A 4B 4C1 4C-1 8A1 8A-1
Size 1 1 2 4 4 4 4 4 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 2A 2B 2F 2F 4A 4A
Type
64.32.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
64.32.1b R 1 1 1 −1 1 1 −1 1 −1 −1 −1 1 1
64.32.1c R 1 1 1 −1 1 1 −1 1 −1 1 1 −1 −1
64.32.1d R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
64.32.1e1 C 1 1 1 −1 1 −1 −1 −1 1 −i i i −i
64.32.1e2 C 1 1 1 −1 1 −1 −1 −1 1 i −i −i i
64.32.1f1 C 1 1 1 1 1 −1 1 −1 −1 −i i −i i
64.32.1f2 C 1 1 1 1 1 −1 1 −1 −1 i −i i −i
64.32.2a R 2 2 2 0 −2 −2 0 2 0 0 0 0 0
64.32.2b R 2 2 2 0 −2 2 0 −2 0 0 0 0 0
64.32.4a R 4 4 −4 0 0 0 0 0 0 0 0 0 0
64.32.4b R 4 −4 0 −2 0 0 2 0 0 0 0 0 0
64.32.4c R 4 −4 0 2 0 0 −2 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{8} - t x^{6} + \left(3 t - 30\right) x^{4} + \left(t + 40\right) x^{2} + \left(-3 t + 5\right)$ Copy content Toggle raw display