Properties

Label 24T22
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $\GL(2,3)$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $22$
Group :  $\GL(2,3)$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,12,17,15,2,11,18,16)(3,10,7,24,4,9,8,23)(5,20,21,13,6,19,22,14), (1,6,14,2,5,13)(3,17,9,4,18,10)(7,11,22,8,12,21)(15,23,19,16,24,20)
$|\Aut(F/K)|$:  $2$

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 3: $S_3$

Degree 4: $S_4$

Degree 6: $S_4$

Degree 8: $\textrm{GL(2,3)}$ x 2

Degree 12: $S_4$

Low degree siblings

8T23 x 2, 16T66

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $12$ $2$ $( 3,21)( 4,22)( 5, 7)( 6, 8)( 9,20)(10,19)(11,16)(12,15)(13,24)(14,23)(17,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $8$ $3$ $( 1, 3,21)( 2, 4,22)( 5,12,20)( 6,11,19)( 7, 9,15)( 8,10,16)(13,23,18) (14,24,17)$
$ 6, 6, 6, 6 $ $8$ $6$ $( 1, 4,21, 2, 3,22)( 5,11,20, 6,12,19)( 7,10,15, 8, 9,16)(13,24,18,14,23,17)$
$ 8, 8, 8 $ $6$ $8$ $( 1, 5,12,24, 2, 6,11,23)( 3, 8,13,19, 4, 7,14,20)( 9,16,21,18,10,15,22,17)$
$ 8, 8, 8 $ $6$ $8$ $( 1, 6,12,23, 2, 5,11,24)( 3, 7,13,20, 4, 8,14,19)( 9,15,21,17,10,16,22,18)$
$ 4, 4, 4, 4, 4, 4 $ $6$ $4$ $( 1,11, 2,12)( 3,14, 4,13)( 5,23, 6,24)( 7,19, 8,20)( 9,22,10,21)(15,18,16,17)$

Group invariants

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

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

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

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