Galois group: 24T114

• Label: 24T114
• Degree: $24$
• Order: $96$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $D_8:C_6$

Group invariants

Abstract group:		$D_8:C_6$
Order:		$96=2^{5} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$114$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,18,6,22,9,13)(2,17,5,21,10,14)(3,19,7,23,11,16)(4,20,8,24,12,15)$, $(1,23,11,22,10,20,8,17,6,16,3,13,2,24,12,21,9,19,7,18,5,15,4,14)$, $(1,14,3,16,5,18,8,20,9,21,11,23,2,13,4,15,6,17,7,19,10,22,12,24)$

Low degree resolvents

$\card{(G/N)}$	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
$32$:	$Z_8 : Z_8^\times$
$48$:	24T38

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 4: $D_{4}$

Degree 6: $C_6$

Degree 8: $Z_8 : Z_8^\times$

Degree 12: $D_4 \times C_3$

Low degree siblings

24T114

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{24}$	$1$	$1$	$0$	$()$
2A	$2^{12}$	$1$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2B	$2^{6},1^{12}$	$2$	$2$	$6$	$(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
2C	$2^{9},1^{6}$	$4$	$2$	$9$	$( 3, 4)( 7, 8)(11,12)(13,20)(14,19)(15,22)(16,21)(17,23)(18,24)$
2D	$2^{9},1^{6}$	$4$	$2$	$9$	$( 3, 4)( 7, 8)(11,12)(13,19)(14,20)(15,21)(16,22)(17,24)(18,23)$
2E	$2^{12}$	$4$	$2$	$12$	$( 1,22)( 2,21)( 3,23)( 4,24)( 5,14)( 6,13)( 7,16)( 8,15)( 9,18)(10,17)(11,19)(12,20)$
3A1	$3^{8}$	$1$	$3$	$16$	$( 1, 9, 6)( 2,10, 5)( 3,11, 7)( 4,12, 8)(13,22,18)(14,21,17)(15,24,20)(16,23,19)$
3A-1	$3^{8}$	$1$	$3$	$16$	$( 1, 6, 9)( 2, 5,10)( 3, 7,11)( 4, 8,12)(13,18,22)(14,17,21)(15,20,24)(16,19,23)$
4A	$4^{6}$	$2$	$4$	$18$	$( 1, 8, 2, 7)( 3, 9, 4,10)( 5,11, 6,12)(13,20,14,19)(15,21,16,22)(17,23,18,24)$
4B	$4^{6}$	$2$	$4$	$18$	$( 1, 8, 2, 7)( 3, 9, 4,10)( 5,11, 6,12)(13,19,14,20)(15,22,16,21)(17,24,18,23)$
4C	$4^{6}$	$4$	$4$	$18$	$( 1,21, 2,22)( 3,24, 4,23)( 5,13, 6,14)( 7,15, 8,16)( 9,17,10,18)(11,20,12,19)$
6A1	$6^{4}$	$1$	$6$	$20$	$( 1,10, 6, 2, 9, 5)( 3,12, 7, 4,11, 8)(13,21,18,14,22,17)(15,23,20,16,24,19)$
6A-1	$6^{4}$	$1$	$6$	$20$	$( 1, 5, 9, 2, 6,10)( 3, 8,11, 4, 7,12)(13,17,22,14,18,21)(15,19,24,16,20,23)$
6B1	$6^{2},3^{4}$	$2$	$6$	$18$	$( 1, 6, 9)( 2, 5,10)( 3, 7,11)( 4, 8,12)(13,17,22,14,18,21)(15,19,24,16,20,23)$
6B-1	$6^{2},3^{4}$	$2$	$6$	$18$	$( 1,10, 6, 2, 9, 5)( 3,12, 7, 4,11, 8)(13,22,18)(14,21,17)(15,24,20)(16,23,19)$
6C1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1, 6, 9)( 2, 5,10)( 3, 8,11, 4, 7,12)(13,24,22,20,18,15)(14,23,21,19,17,16)$
6C-1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1,10, 6, 2, 9, 5)( 3,11, 7)( 4,12, 8)(13,16,18,19,22,23)(14,15,17,20,21,24)$
6D1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1, 6, 9)( 2, 5,10)( 3, 8,11, 4, 7,12)(13,23,22,19,18,16)(14,24,21,20,17,15)$
6D-1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1,10, 6, 2, 9, 5)( 3,11, 7)( 4,12, 8)(13,15,18,20,22,24)(14,16,17,19,21,23)$
6E1	$6^{4}$	$4$	$6$	$20$	$( 1,18, 6,22, 9,13)( 2,17, 5,21,10,14)( 3,19, 7,23,11,16)( 4,20, 8,24,12,15)$
6E-1	$6^{4}$	$4$	$6$	$20$	$( 1,14, 9,21, 6,17)( 2,13,10,22, 5,18)( 3,15,11,24, 7,20)( 4,16,12,23, 8,19)$
8A	$8^{3}$	$4$	$8$	$21$	$( 1,15, 8,22, 2,16, 7,21)( 3,17, 9,24, 4,18,10,23)( 5,19,11,14, 6,20,12,13)$
8B	$8^{3}$	$4$	$8$	$21$	$( 1,16, 7,22, 2,15, 8,21)( 3,18,10,24, 4,17, 9,23)( 5,20,12,14, 6,19,11,13)$
12A1	$12^{2}$	$2$	$12$	$22$	$( 1,11,10, 8, 6, 3, 2,12, 9, 7, 5, 4)(13,23,21,20,18,16,14,24,22,19,17,15)$
12A-1	$12^{2}$	$2$	$12$	$22$	$( 1, 3, 5, 8, 9,11, 2, 4, 6, 7,10,12)(13,16,17,20,22,23,14,15,18,19,21,24)$
12B1	$12^{2}$	$2$	$12$	$22$	$( 1,11,10, 8, 6, 3, 2,12, 9, 7, 5, 4)(13,24,21,19,18,15,14,23,22,20,17,16)$
12B-1	$12^{2}$	$2$	$12$	$22$	$( 1, 4, 5, 7, 9,12, 2, 3, 6, 8,10,11)(13,16,17,20,22,23,14,15,18,19,21,24)$
12C1	$12^{2}$	$4$	$12$	$22$	$( 1,18, 5,21, 9,13, 2,17, 6,22,10,14)( 3,19, 8,24,11,16, 4,20, 7,23,12,15)$
12C-1	$12^{2}$	$4$	$12$	$22$	$( 1,14,10,22, 6,17, 2,13, 9,21, 5,18)( 3,15,12,23, 7,20, 4,16,11,24, 8,19)$
24A1	$24$	$4$	$24$	$23$	$( 1,18,11,15,10,14, 8,23, 6,22, 3,20, 2,17,12,16, 9,13, 7,24, 5,21, 4,19)$
24A-1	$24$	$4$	$24$	$23$	$( 1,14, 3,16, 5,18, 8,20, 9,21,11,23, 2,13, 4,15, 6,17, 7,19,10,22,12,24)$
24B1	$24$	$4$	$24$	$23$	$( 1,18,12,16,10,14, 7,24, 6,22, 4,19, 2,17,11,15, 9,13, 8,23, 5,21, 3,20)$
24B-1	$24$	$4$	$24$	$23$	$( 1,14, 4,15, 5,18, 7,19, 9,21,12,24, 2,13, 3,16, 6,17, 8,20,10,22,11,23)$

Malle's constant $a(G)$:     $1/6$

Character table

33 x 33 character table

Regular extensions

Data not computed