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Properties

Label 24T11
Order \(24\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $11$
Group :  $C_2^2\times S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,8)(2,7)(3,21)(4,22)(5,11)(6,12)(9,15)(10,16)(13,19)(14,20)(17,24)(18,23), (1,7)(2,8)(3,5)(4,6)(9,24)(10,23)(11,21)(12,22)(13,20)(14,19)(15,17)(16,18), (1,6)(2,5)(3,15)(4,16)(7,11)(8,12)(9,21)(10,22)(13,17)(14,18)(19,24)(20,23)
$|\Aut(F/K)|$:  $24$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $C_2^3$
12:  $D_{6}$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 3: $S_3$

Degree 4: $C_2^2$ x 7

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

Degree 8: $C_2^3$

Degree 12: $D_6$ x 3, $S_3 \times C_2^2$ x 4

Low degree siblings

12T10 x 4

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

Group invariants

Order:  $24=2^{3} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [24, 14]
Character table:    
      2  3  3  3  2  2  3  3  3  2  3  2  3
      3  1  .  .  1  1  .  .  1  1  1  1  1

        1a 2a 2b 6a 6b 2c 2d 2e 3a 2f 6c 2g
     2P 1a 1a 1a 3a 3a 1a 1a 1a 3a 1a 3a 1a
     3P 1a 2a 2b 2g 2f 2c 2d 2e 1a 2f 2e 2g
     5P 1a 2a 2b 6a 6b 2c 2d 2e 3a 2f 6c 2g

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