Properties

Label 24T38
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_6\times D_4$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $38$
Group :  $C_6\times D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,13)(2,14)(5,18)(6,17)(9,21)(10,22), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,15,6,19,9,24,13,3,17,7,21,11)(2,16,5,20,10,23,14,4,18,8,22,12)
$|\Aut(F/K)|$:  $12$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
3:  $C_3$
4:  $C_2^2$ x 7
6:  $C_6$ x 7
8:  $D_{4}$ x 2, $C_2^3$
12:  $C_6\times C_2$ x 7
16:  $D_4\times C_2$
24:  $D_4 \times C_3$ x 2, 24T3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $C_3$

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 6: $C_6$ x 3

Degree 8: $D_4\times C_2$

Degree 12: $C_6\times C_2$, $D_4 \times C_3$ x 2

Low degree siblings

24T38 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,15)( 4,16)( 7,19)( 8,20)(11,24)(12,23)$
$ 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 $ $2$ $2$ $( 1, 2)( 3,16)( 4,15)( 5, 6)( 7,20)( 8,19)( 9,10)(11,23)(12,24)(13,14)(17,18) (21,22)$
$ 12, 12 $ $2$ $12$ $( 1, 3, 6, 7, 9,11,13,15,17,19,21,24)( 2, 4, 5, 8,10,12,14,16,18,20,22,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 3,17,19, 9,11)( 2, 4,18,20,10,12)( 5, 8,22,23,14,16)( 6, 7,21,24,13,15)$
$ 12, 12 $ $2$ $12$ $( 1, 4, 6, 8, 9,12,13,16,17,20,21,23)( 2, 3, 5, 7,10,11,14,15,18,19,22,24)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 4,17,20, 9,12)( 2, 3,18,19,10,11)( 5, 7,22,24,14,15)( 6, 8,21,23,13,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1, 5, 9,14,17,22)( 2, 6,10,13,18,21)( 3,20,11, 4,19,12)( 7,23,15, 8,24,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3, 7,11,15,19,24)( 4, 8,12,16,20,23)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 6, 9,13,17,21)( 2, 5,10,14,18,22)( 3,19,11)( 4,20,12)( 7,24,15)( 8,23,16)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,13,19)( 2, 8,14,20)( 3, 9,15,21)( 4,10,16,22)( 5,12,18,23)( 6,11,17,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,21)( 4,22)( 5,12)( 6,11)( 9,15)(10,16)(13,19)(14,20)(17,24) (18,23)$
$ 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,13,20)( 2, 7,14,19)( 3,10,15,22)( 4, 9,16,21)( 5,11,18,24)( 6,12,17,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 8)( 2, 7)( 3,22)( 4,21)( 5,11)( 6,12)( 9,16)(10,15)(13,20)(14,19)(17,23) (18,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 9,17)( 2,10,18)( 3,11,19)( 4,12,20)( 5,14,22)( 6,13,21)( 7,15,24) ( 8,16,23)$
$ 6, 6, 3, 3, 3, 3 $ $2$ $6$ $( 1, 9,17)( 2,10,18)( 3,24,19,15,11, 7)( 4,23,20,16,12, 8)( 5,14,22)( 6,13,21)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,10,17, 2, 9,18)( 3,12,19, 4,11,20)( 5,13,22, 6,14,21)( 7,16,24, 8,15,23)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,10,17, 2, 9,18)( 3,23,19,16,11, 8)( 4,24,20,15,12, 7)( 5,13,22, 6,14,21)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,11, 9,19,17, 3)( 2,12,10,20,18, 4)( 5,16,14,23,22, 8)( 6,15,13,24,21, 7)$
$ 12, 12 $ $2$ $12$ $( 1,11,21, 7,17, 3,13,24, 9,19, 6,15)( 2,12,22, 8,18, 4,14,23,10,20, 5,16)$
$ 6, 6, 6, 6 $ $2$ $6$ $( 1,12, 9,20,17, 4)( 2,11,10,19,18, 3)( 5,15,14,24,22, 7)( 6,16,13,23,21, 8)$
$ 12, 12 $ $2$ $12$ $( 1,12,21, 8,17, 4,13,23, 9,20, 6,16)( 2,11,22, 7,18, 3,14,24,10,19, 5,15)$
$ 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 $ $1$ $2$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5,17)( 6,18)( 7,20)( 8,19)( 9,22)(10,21)(11,23) (12,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15) ( 8,23,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,21,17,13, 9, 6)( 2,22,18,14,10, 5)( 3,24,19,15,11, 7)( 4,23,20,16,12, 8)$
$ 6, 6, 6, 6 $ $1$ $6$ $( 1,22,17,14, 9, 5)( 2,21,18,13,10, 6)( 3,23,19,16,11, 8)( 4,24,20,15,12, 7)$

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 45]
Character table: Data not available.