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Properties

Label 8T28
Order \(64\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(((C_4 \times C_2): C_2):C_2):C_2$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $28$
Group :  $(((C_4 \times C_2): C_2):C_2):C_2$
CHM label :  $1/2[2^{4}]dD(4)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,2,3,4,5,6,7,8), (2,6)(3,7), (1,3)(5,7)
$|\Aut(F/K)|$:  $2$

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

8T27 x 2, 8T28, 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

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 32]
Character table:    
      2  6  4  4  3  5  3  3  3  3  4  4  4  6

        1a 2a 2b 4a 2c 4b 4c 8a 8b 2d 4d 2e 2f
     2P 1a 1a 1a 2c 1a 2a 2a 4d 4d 1a 2f 1a 1a
     3P 1a 2a 2b 4a 2c 4c 4b 8b 8a 2d 4d 2e 2f
     5P 1a 2a 2b 4a 2c 4b 4c 8a 8b 2d 4d 2e 2f
     7P 1a 2a 2b 4a 2c 4c 4b 8b 8a 2d 4d 2e 2f

X.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
X.3      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
X.5      1 -1 -1  1  1  A -A -A  A  1 -1 -1  1
X.6      1 -1 -1  1  1 -A  A  A -A  1 -1 -1  1
X.7      1 -1  1 -1  1  A -A  A -A  1 -1  1  1
X.8      1 -1  1 -1  1 -A  A -A  A  1 -1  1  1
X.9      2 -2  .  .  2  .  .  .  . -2  2  .  2
X.10     2  2  .  .  2  .  .  .  . -2 -2  .  2
X.11     4  .  .  . -4  .  .  .  .  .  .  .  4
X.12     4  . -2  .  .  .  .  .  .  .  .  2 -4
X.13     4  .  2  .  .  .  .  .  .  .  . -2 -4

A = -E(4)
  = -Sqrt(-1) = -i