↑

Properties

Label 8T48
Order \(1344\)
n \(8\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $C_2^3:\GL(3,2)$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $48$
Group :  $C_2^3:\GL(3,2)$
CHM label :  $E(8):L_{7}=AL(8)$
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), (1,2)(5,6), (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)
168:  $\GL(3,2)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Low degree siblings

8T48, 14T34 x 2, 28T153, 28T159 x 2, 42T210 x 2, 42T211 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, 2, 2, 2 $ $7$ $2$ $(1,3)(2,8)(4,6)(5,7)$
$ 2, 2, 1, 1, 1, 1 $ $42$ $2$ $(4,5)(6,7)$
$ 2, 2, 2, 2 $ $42$ $2$ $(1,3)(2,8)(4,7)(5,6)$
$ 4, 4 $ $84$ $4$ $(1,5,8,4)(2,6,3,7)$
$ 3, 3, 1, 1 $ $224$ $3$ $(3,4,5)(6,8,7)$
$ 6, 2 $ $224$ $6$ $(1,3,6,2,8,5)(4,7)$
$ 4, 2, 1, 1 $ $168$ $4$ $(3,4,8,7)(5,6)$
$ 4, 4 $ $168$ $4$ $(1,3,6,7)(2,8,5,4)$
$ 7, 1 $ $192$ $7$ $(2,3,4,7,5,8,6)$
$ 7, 1 $ $192$ $7$ $(2,3,4,8,6,5,7)$

Group invariants

Order:  $1344=2^{6} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1344, 11686]
Character table:    
      2  6  6  5  5  4  1  1  3  3  .  .
      3  1  1  .  .  .  1  1  .  .  .  .
      7  1  .  .  .  .  .  .  .  .  1  1

        1a 2a 2b 2c 4a 3a 6a 4b 4c 7a 7b
     2P 1a 1a 1a 1a 2a 3a 3a 2b 2c 7a 7b
     3P 1a 2a 2b 2c 4a 1a 2a 4b 4c 7b 7a
     5P 1a 2a 2b 2c 4a 3a 6a 4b 4c 7b 7a
     7P 1a 2a 2b 2c 4a 3a 6a 4b 4c 1a 1a

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      3  3 -1 -1 -1  .  .  1  1  A /A
X.3      3  3 -1 -1 -1  .  .  1  1 /A  A
X.4      6  6  2  2  2  .  .  .  . -1 -1
X.5      7 -1  3 -1 -1  1 -1  1 -1  .  .
X.6      7  7 -1 -1 -1  1  1 -1 -1  .  .
X.7      7 -1 -1  3 -1  1 -1 -1  1  .  .
X.8      8  8  .  .  . -1 -1  .  .  1  1
X.9     14 -2  2  2 -2 -1  1  .  .  .  .
X.10    21 -3  1 -3  1  .  . -1  1  .  .
X.11    21 -3 -3  1  1  .  .  1 -1  .  .

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