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Properties

Label 8T29
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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Show commands: Magma

magma: G := TransitiveGroup(8, 29);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(((C_4 \times C_2): C_2):C_2):C_2$
CHM label:   $E(8):D_{8}=[2^{3}]D(4)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3)(2,8)(4,6)(5,7), (1,8)(2,3)(4,5)(6,7), (1,5)(2,6)(3,7)(4,8), (1,3)(4,5,6,7), (1,3)(5,7)
magma: Generators(G);
 

Low degree resolvents

|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

8T29 x 5, 8T31 x 2, 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1 $ $4$ $2$ $(4,5)(6,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(4,6)(5,7)$
$ 2, 2, 1, 1, 1, 1 $ $4$ $2$ $(2,8)(5,7)$
$ 4, 2, 1, 1 $ $8$ $4$ $(2,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,5)(6,7)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,2)(3,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $2$ $2$ $(1,2)(3,8)(4,7)(5,6)$
$ 4, 4 $ $4$ $4$ $(1,2,3,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,5)(3,6)(7,8)$
$ 4, 4 $ $8$ $4$ $(1,4,2,5)(3,6,8,7)$
$ 4, 4 $ $4$ $4$ $(1,4,3,6)(2,5,8,7)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,4)(2,7)(3,6)(5,8)$
$ 4, 4 $ $8$ $4$ $(1,4,2,7)(3,6,8,5)$
$ 4, 4 $ $4$ $4$ $(1,4,3,6)(2,7,8,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $64=2^{6}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  64.138
magma: IdentifyGroup(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 −2 0 0 0 0 2 0 0 0 0 0
64.138.2d R 2 2 −2 2 −2 2 0 0 0 0 −2 0 0 0 0 0
64.138.2e R 2 2 2 −2 −2 0 0 0 −2 0 0 2 0 0 0 0
64.138.2f R 2 2 2 −2 −2 0 0 0 2 0 0 −2 0 0 0 0
64.138.4a R 4 −4 0 0 0 0 0 −2 0 2 0 0 0 0 0 0
64.138.4b R 4 −4 0 0 0 0 0 2 0 −2 0 0 0 0 0 0

magma: CharacterTable(G);