Galois group: 24T42

• Label: 24T42
• Degree: $24$
• Order: $48$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3:\SD_{16}$

Group invariants

Abstract group:		$C_3:\SD_{16}$
Order:		$48=2^{4} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$42$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$6$
Generators:		$(1,15,2,16)(3,13,4,14)(5,11,6,12)(7,10,8,9)(17,24,18,23)(19,21,20,22)$, $(1,18,9,2,17,10)(3,7,11,15,19,24)(4,8,12,16,20,23)(5,22,14)(6,21,13)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$
$8$:	$D_{4}$
$12$:	$D_{6}$
$16$:	$QD_{16}$
$24$:	$(C_6\times C_2):C_2$

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: $QD_{16}$

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

Low degree siblings

There are no siblings with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

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	$2^{9},1^{6}$	$4$	$2$	$9$	$( 3,16)( 4,15)( 5, 6)( 7,20)( 8,19)(11,23)(12,24)(13,14)(21,22)$
3A	$3^{8}$	$2$	$3$	$16$	$( 1,17, 9)( 2,18,10)( 3,19,11)( 4,20,12)( 5,22,14)( 6,21,13)( 7,24,15)( 8,23,16)$
4A	$4^{6}$	$2$	$4$	$18$	$( 1,13, 2,14)( 3,16, 4,15)( 5,17, 6,18)( 7,19, 8,20)( 9,21,10,22)(11,23,12,24)$
4B	$4^{6}$	$12$	$4$	$18$	$( 1, 3, 2, 4)( 5,23, 6,24)( 7,22, 8,21)( 9,19,10,20)(11,18,12,17)(13,15,14,16)$
6A	$6^{4}$	$2$	$6$	$20$	$( 1,18, 9, 2,17,10)( 3,20,11, 4,19,12)( 5,21,14, 6,22,13)( 7,23,15, 8,24,16)$
6B1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1, 9,17)( 2,10,18)( 3,23,19,16,11, 8)( 4,24,20,15,12, 7)( 5,13,22, 6,14,21)$
6B-1	$6^{3},3^{2}$	$4$	$6$	$19$	$( 1,17, 9)( 2,18,10)( 3, 8,11,16,19,23)( 4, 7,12,15,20,24)( 5,21,14, 6,22,13)$
8A1	$8^{3}$	$6$	$8$	$21$	$( 1, 4,13,15, 2, 3,14,16)( 5,23,17,12, 6,24,18,11)( 7,10,19,22, 8, 9,20,21)$
8A-1	$8^{3}$	$6$	$8$	$21$	$( 1, 3,13,16, 2, 4,14,15)( 5,24,17,11, 6,23,18,12)( 7, 9,19,21, 8,10,20,22)$
12A	$12^{2}$	$4$	$12$	$22$	$( 1,21,18,14, 9, 6, 2,22,17,13,10, 5)( 3,23,20,15,11, 8, 4,24,19,16,12, 7)$

Malle's constant $a(G)$:     $1/9$

Character table

		1A	2A	2B	3A	4A	4B	6A	6B1	6B-1	8A1	8A-1	12A
	Size	1	1	4	2	2	12	2	4	4	6	6	4
	2 P	1A	1A	1A	3A	2A	2A	3A	3A	3A	4A	4A	6A
	3 P	1A	2A	2B	1A	4A	4B	2A	2B	2B	8A1	8A-1	4A
	Type
48.16.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
48.16.1b	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$
48.16.1c	R	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
48.16.1d	R	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$
48.16.2a	R	$2$	$2$	$2$	$-1$	$2$	$0$	$-1$	$-1$	$-1$	$0$	$0$	$-1$
48.16.2b	R	$2$	$2$	$0$	$2$	$-2$	$0$	$2$	$0$	$0$	$0$	$0$	$-2$
48.16.2c	R	$2$	$2$	$-2$	$-1$	$2$	$0$	$-1$	$1$	$1$	$0$	$0$	$-1$
48.16.2d1	C	$2$	$-2$	$0$	$2$	$0$	$0$	$-2$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$0$
48.16.2d2	C	$2$	$-2$	$0$	$2$	$0$	$0$	$-2$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$0$
48.16.2e1	C	$2$	$2$	$0$	$-1$	$-2$	$0$	$-1$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$1$
48.16.2e2	C	$2$	$2$	$0$	$-1$	$-2$	$0$	$-1$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$1$
48.16.4a	S	$4$	$-4$	$0$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed