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Properties

Label 8T30
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, 30);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $30$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(((C_4 \times C_2): C_2):C_2):C_2$
CHM label:   $1/2[2^{4}]cD(4)$
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:  (1,2,3,8)(4,5,6,7), (2,6)(3,7), (1,3)(4,8)(5,7)
magma: Generators(G);
 

Low degree resolvents

|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: $D_{4}$

Low degree siblings

8T30 x 3, 16T143 x 2, 16T167 x 2, 16T168 x 2, 16T169 x 2, 32T157 x 2, 32T177, 32T178

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 $ $4$ $2$ $(3,7)(4,8)$
$ 4, 1, 1, 1, 1 $ $4$ $4$ $(2,4,6,8)$
$ 2, 2, 2, 1, 1 $ $8$ $2$ $(2,4)(3,7)(6,8)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 2, 2 $ $8$ $4$ $(1,2)(3,8,7,4)(5,6)$
$ 4, 4 $ $8$ $4$ $(1,2,3,4)(5,6,7,8)$
$ 4, 4 $ $8$ $4$ $(1,2,3,8)(4,5,6,7)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 4 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 4, 2, 2 $ $4$ $4$ $(1,3,5,7)(2,6)(4,8)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

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:  $4$
Label:  64.34
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C 4D1 4D-1 4E1 4E-1
Size 1 1 2 4 4 8 4 4 4 8 8 8 8
2 P 1A 1A 1A 1A 1A 1A 2B 2A 2B 2C 2C 2D 2D
Type
64.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
64.34.1b R 1 1 1 1 1 −1 −1 1 −1 −1 −1 1 1
64.34.1c R 1 1 1 1 1 −1 −1 1 −1 1 1 −1 −1
64.34.1d R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
64.34.1e1 C 1 1 1 −1 −1 −1 1 1 1 −i i i −i
64.34.1e2 C 1 1 1 −1 −1 −1 1 1 1 i −i −i i
64.34.1f1 C 1 1 1 −1 −1 1 −1 1 −1 −i i −i i
64.34.1f2 C 1 1 1 −1 −1 1 −1 1 −1 i −i i −i
64.34.2a R 2 2 2 −2 2 0 0 −2 0 0 0 0 0
64.34.2b R 2 2 2 2 −2 0 0 −2 0 0 0 0 0
64.34.4a R 4 4 −4 0 0 0 0 0 0 0 0 0 0
64.34.4b R 4 −4 0 0 0 0 −2 0 2 0 0 0 0
64.34.4c R 4 −4 0 0 0 0 2 0 −2 0 0 0 0

magma: CharacterTable(G);