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Properties

Label 24T118
Order \(96\)
n \(24\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3:D_8:C_2$

Related objects

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Group action invariants

Degree $n$ :  $24$
Transitive number $t$ :  $118$
Group :  $C_3:D_8:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,6)(2,10,5)(3,11,7)(4,12,8)(13,21,18,14,22,17)(15,23,20,16,24,19), (1,18,8,23,2,17,7,24)(3,15,9,22,4,16,10,21)(5,14,11,20,6,13,12,19), (1,19)(2,20)(3,17)(4,18)(5,15)(6,16)(7,14)(8,13)(9,23)(10,24)(11,21)(12,22)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
6:  $S_3$
8:  $D_{4}$ x 2, $C_2^3$
12:  $D_{6}$ x 3
16:  $D_4\times C_2$
24:  $S_3 \times C_2^2$, $(C_6\times C_2):C_2$ x 2
32:  $Z_8 : Z_8^\times$
48:  24T25

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 4: $D_{4}$

Degree 6: $S_3$

Degree 8: $Z_8 : Z_8^\times$

Degree 12: $(C_6\times C_2):C_2$

Low degree siblings

24T118

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $96=2^{5} \cdot 3$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [96, 139]
Character table: Data not available.