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Properties

Label 8T35
Degree $8$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2 \wr C_2\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(8, 35);
 

Group action invariants

Degree $n$:  $8$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $35$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \wr C_2\wr C_2$
CHM label:  $[2^{4}]D(4)$
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:  (1,2,3,8)(4,5,6,7), (4,8), (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
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Low degree siblings

8T35 x 7, 16T376 x 4, 16T388 x 4, 16T390 x 4, 16T391 x 4, 16T393 x 4, 16T395 x 4, 16T396 x 4, 16T401 x 4, 32T852 x 4, 32T853 x 2, 32T854 x 2, 32T872 x 2, 32T876 x 4, 32T877 x 2, 32T880 x 2, 32T882 x 2, 32T883 x 4, 32T884 x 2, 32T885 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $4$ $2$ $(3,7)(4,8)$
$ 2, 2, 1, 1, 1, 1 $ $4$ $2$ $(2,4)(6,8)$
$ 4, 1, 1, 1, 1 $ $4$ $4$ $(2,4,6,8)$
$ 2, 2, 2, 1, 1 $ $8$ $2$ $(2,4)(3,7)(6,8)$
$ 4, 2, 1, 1 $ $8$ $4$ $(2,4,6,8)(3,7)$
$ 2, 2, 1, 1, 1, 1 $ $2$ $2$ $(2,6)(4,8)$
$ 2, 2, 2, 1, 1 $ $4$ $2$ $(2,6)(3,7)(4,8)$
$ 2, 2, 2, 2 $ $8$ $2$ $(1,2)(3,4)(5,6)(7,8)$
$ 4, 2, 2 $ $16$ $4$ $(1,2)(3,4,7,8)(5,6)$
$ 4, 4 $ $16$ $4$ $(1,2,3,4)(5,6,7,8)$
$ 8 $ $16$ $8$ $(1,2,3,4,5,6,7,8)$
$ 4, 4 $ $8$ $4$ $(1,2,5,6)(3,4,7,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,4)(5,7)(6,8)$
$ 4, 2, 2 $ $8$ $4$ $(1,3)(2,4,6,8)(5,7)$
$ 4, 4 $ $4$ $4$ $(1,3,5,7)(2,4,6,8)$
$ 2, 2, 2, 2 $ $4$ $2$ $(1,3)(2,6)(4,8)(5,7)$
$ 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:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $4$
Label:  128.928
magma: IdentifyGroup(G);
 
Character table:    
      2  7  5  5  5  5  4  4  6  5  4  3  3  3  4  5  4  5  5  5  7

        1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     2P 1a 1a 1a 1a 2e 1a 2e 1a 1a 1a 2b 2h 4g 2j 1a 2e 2j 1a 2e 1a
     3P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     5P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j
     7P 1a 2a 2b 2c 4a 2d 4b 2e 2f 2g 4c 4d 8a 4e 2h 4f 4g 2i 4h 2j

X.1      1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1 -1  1  1
X.3      1 -1  1 -1  1  1 -1  1 -1  1 -1 -1  1  1  1 -1  1 -1  1  1
X.4      1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1
X.5      1 -1  1  1 -1 -1  1  1 -1  1 -1  1 -1  1  1 -1  1  1 -1  1
X.6      1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1  1  1  1 -1 -1  1
X.7      1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1  1  1 -1 -1  1
X.8      1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1  1  1  1  1  1  1
X.9      2  2  2  .  .  .  .  2  2  .  .  .  .  . -2 -2 -2  .  .  2
X.10     2 -2  2  .  .  .  .  2 -2  .  .  .  .  . -2  2 -2  .  .  2
X.11     2  . -2  2  .  . -2  2  .  .  .  .  .  .  2  . -2  2  .  2
X.12     2  . -2 -2  .  .  2  2  .  .  .  .  .  .  2  . -2 -2  .  2
X.13     2  . -2  . -2  2  .  2  .  .  .  .  .  . -2  .  2  . -2  2
X.14     2  . -2  .  2 -2  .  2  .  .  .  .  .  . -2  .  2  .  2  2
X.15     4  .  .  .  .  .  . -4  . -2  .  .  .  2  .  .  .  .  .  4
X.16     4  .  .  .  .  .  . -4  .  2  .  .  . -2  .  .  .  .  .  4
X.17     4 -2  . -2  2  .  .  .  2  .  .  .  .  .  .  .  .  2 -2 -4
X.18     4 -2  .  2 -2  .  .  .  2  .  .  .  .  .  .  .  . -2  2 -4
X.19     4  2  . -2 -2  .  .  . -2  .  .  .  .  .  .  .  .  2  2 -4
X.20     4  2  .  2  2  .  .  . -2  .  .  .  .  .  .  .  . -2 -2 -4

magma: CharacterTable(G);