Properties

Label 24T30
Order \(48\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^3\times S_3$

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $30$
Group :  $C_2^3\times S_3$
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), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24), (1,23)(2,24)(3,21)(4,22)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,13)(12,14), (1,20)(2,19)(3,10)(4,9)(5,23)(6,24)(7,13)(8,14)(11,18)(12,17)(15,21)(16,22)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
6:  $S_3$
8:  $C_2^3$ x 15
12:  $D_{6}$ x 7
16:  $C_2^4$
24:  $S_3 \times C_2^2$ x 7

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: $D_{6}$ x 7

Degree 8: $C_2^3$

Degree 12: $S_3 \times C_2^2$ x 7

Low degree siblings

24T30 x 7

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

Group invariants

Order:  $48=2^{4} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [48, 51]
Character table: Data not available.