Properties

Label 24T27
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_4\times S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $27$
Group :  $C_2\times C_4\times S_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,11)(4,12)(5,22)(6,21)(9,17)(10,18)(15,24)(16,23), (1,23,2,24)(3,21,4,22)(5,19,6,20)(7,17,8,18)(9,16,10,15)(11,13,12,14), (1,20,2,19)(3,9,4,10)(5,24,6,23)(7,13,8,14)(11,17,12,18)(15,21,16,22)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
6:  $S_3$
8:  $C_4\times C_2$ x 6, $C_2^3$
12:  $D_{6}$ x 3
16:  $C_4\times C_2^2$
24:  $S_3 \times C_2^2$, $S_3 \times C_4$ 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: $D_{6}$ x 3

Degree 8: $C_4\times C_2$

Degree 12: $S_3 \times C_2^2$, $S_3 \times C_4$ x 2

Low degree siblings

24T27 x 3

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 35]
Character table: Data not available.