Properties

Label 24T32
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_8\times S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $32$
Group :  $C_8\times S_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,2,13)(3,23,4,24)(5,10,6,9)(7,19,8,20)(11,16,12,15)(17,22,18,21), (1,12,13,23,2,11,14,24)(3,22,15,9,4,21,16,10)(5,7,17,20,6,8,18,19)
$|\Aut(F/K)|$:  $8$

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:  $C_8$ x 2, $C_4\times C_2$
12:  $D_{6}$
16:  $C_8\times C_2$
24:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $C_4$

Degree 6: $D_{6}$

Degree 8: $C_8$

Degree 12: $S_3 \times C_4$

Low degree siblings

24T32

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 4]
Character table: Data not available.