Properties

Label 24T65
Degree $24$
Order $72$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{12}\times S_3$

Related objects

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Group action invariants

Degree $n$:  $24$
Transitive number $t$:  $65$
Group:  $C_{12}\times S_3$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $12$
Generators:  (1,22,17,13,9,5,2,21,18,14,10,6)(3,16,4,15)(7,20,8,19)(11,24,12,23), (1,23,17,16,9,7,2,24,18,15,10,8)(3,21,19,14,11,6,4,22,20,13,12,5)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$8$:  $C_4\times C_2$
$12$:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
$18$:  $S_3\times C_3$
$24$:  $S_3 \times C_4$, 24T2
$36$:  $C_6\times S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

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

Degree 6: $S_3\times C_3$

Degree 8: $C_4\times C_2$

Degree 12: $C_6\times S_3$

Low degree siblings

36T27 x 2

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

Group invariants

Order:  $72=2^{3} \cdot 3^{2}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [72, 27]
Character table: not available.