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Properties

Label 8T23
Order \(48\)
n \(8\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\textrm{GL(2,3)}$

Related objects

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Group action invariants

Degree $n$ :  $8$
Transitive number $t$ :  $23$
Group :  $\textrm{GL(2,3)}$
CHM label :  $2S_{4}(8)=GL(2,3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8), (1,3,8)(4,5,7)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$
24:  $S_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $S_4$

Low degree siblings

8T23, 16T66, 24T22

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 1, 1 $ $12$ $2$ $(2,3)(4,8)(6,7)$
$ 3, 3, 1, 1 $ $8$ $3$ $(2,7,8)(3,4,6)$
$ 8 $ $6$ $8$ $(1,2,3,4,5,6,7,8)$
$ 4, 4 $ $6$ $4$ $(1,2,5,6)(3,8,7,4)$
$ 6, 2 $ $8$ $6$ $(1,2,7,5,6,3)(4,8)$
$ 8 $ $6$ $8$ $(1,2,8,3,5,6,4,7)$
$ 2, 2, 2, 2 $ $1$ $2$ $(1,5)(2,6)(3,7)(4,8)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 29]
Character table:    
     2  4  2  1  3  3  1  3  4
     3  1  .  1  .  .  1  .  1

       1a 2a 3a 8a 4a 6a 8b 2b
    2P 1a 1a 3a 4a 2b 3a 4a 1a
    3P 1a 2a 1a 8a 4a 2b 8b 2b
    5P 1a 2a 3a 8b 4a 6a 8a 2b
    7P 1a 2a 3a 8b 4a 6a 8a 2b

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

A = E(8)+E(8)^3
  = Sqrt(-2) = i2