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Properties

Label 24T6
Degree $24$
Order $24$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_3:C_4$

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Group action invariants

Degree $n$:  $24$
Transitive number $t$:  $6$
Group:  $C_2\times C_3:C_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $24$
Generators:  (1,7,17,24,9,15)(2,8,18,23,10,16)(3,13,19,6,11,21)(4,14,20,5,12,22), (1,22,2,21)(3,15,4,16)(5,18,6,17)(7,12,8,11)(9,14,10,13)(19,24,20,23)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_3 : C_4$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$

Degree 4: $C_4$ x 2, $C_2^2$

Degree 6: $S_3$, $D_{6}$ x 2

Degree 8: $C_4\times C_2$

Degree 12: $D_6$, $C_3 : C_4$ x 2

Low degree siblings

There are no siblings 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, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 3, 2, 4)( 5,24, 6,23)( 7,21, 8,22)( 9,19,10,20)(11,18,12,17)(13,16,14,15)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 4, 2, 3)( 5,23, 6,24)( 7,22, 8,21)( 9,20,10,19)(11,17,12,18)(13,15,14,16)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 5, 2, 6)( 3,24, 4,23)( 7,20, 8,19)( 9,22,10,21)(11,15,12,16)(13,17,14,18)$
$ 4, 4, 4, 4, 4, 4 $ $3$ $4$ $( 1, 6, 2, 5)( 3,23, 4,24)( 7,19, 8,20)( 9,21,10,22)(11,16,12,15)(13,18,14,17)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 7,17,24, 9,15)( 2, 8,18,23,10,16)( 3,13,19, 6,11,21)( 4,14,20, 5,12,22)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 8,17,23, 9,16)( 2, 7,18,24,10,15)( 3,14,19, 5,11,22)( 4,13,20, 6,12,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,23)( 2,24)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,18)(16,17)(19,22) (20,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,24)( 2,23)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,17)(16,18)(19,21) (20,22)$

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [24, 7]
Character table:    
      2  3  3  3  3  3  3  2  2  2  2  3  3
      3  1  1  .  .  .  .  1  1  1  1  1  1

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

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

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