Properties

Label 8T36
Order \(168\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_2^3:(C_7: C_3)$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $36$
Group :  $C_2^3:(C_7: C_3)$
CHM label :  $E(8):F_{21}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3)(2,8)(4,6)(5,7), (1,2,6,3,4,5,7), (1,8)(2,3)(4,5)(6,7), (1,2,3)(4,6,5), (1,5)(2,6)(3,7)(4,8)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

14T11, 24T283, 28T27, 42T26

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 $ $1$ $1$ $()$
$ 3, 3, 1, 1 $ $28$ $3$ $(3,6,7)(4,8,5)$
$ 3, 3, 1, 1 $ $28$ $3$ $(3,7,6)(4,5,8)$
$ 7, 1 $ $24$ $7$ $(2,3,6,5,7,8,4)$
$ 7, 1 $ $24$ $7$ $(2,4,8,7,5,6,3)$
$ 6, 2 $ $28$ $6$ $(1,2)(3,4,6,8,7,5)$
$ 6, 2 $ $28$ $6$ $(1,2)(3,5,7,8,6,4)$
$ 2, 2, 2, 2 $ $7$ $2$ $(1,2)(3,8)(4,7)(5,6)$

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 43]
Character table:   
     2  3  1  1  .  .   1   1  3
     3  1  1  1  .  .   1   1  1
     7  1  .  .  1  1   .   .  .

       1a 3a 3b 7a 7b  6a  6b 2a
    2P 1a 3b 3a 7a 7b  3a  3b 1a
    3P 1a 1a 1a 7b 7a  2a  2a 2a
    5P 1a 3b 3a 7b 7a  6b  6a 2a
    7P 1a 3a 3b 1a 1a  6a  6b 2a

X.1     1  1  1  1  1   1   1  1
X.2     1  A /A  1  1  /A   A  1
X.3     1 /A  A  1  1   A  /A  1
X.4     3  .  .  B /B   .   .  3
X.5     3  .  . /B  B   .   .  3
X.6     7  1  1  .  .  -1  -1 -1
X.7     7  A /A  .  . -/A  -A -1
X.8     7 /A  A  .  .  -A -/A -1

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7