Galois group: 24T63

• Label: 24T63
• Degree: $24$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:C_8$

Group invariants

Abstract group:		$C_3^2:C_8$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$24$
Transitive number $t$:		$63$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$12$
Generators:		$(1,15,22,20,2,16,21,19)(3,10,24,6,4,9,23,5)(7,14,11,18,8,13,12,17)$, $(1,17,10)(2,18,9)(5,13,21)(6,14,22)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$8$:	$C_8$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: $C_4$

Degree 6: $C_3^2:C_4$

Degree 8: $C_8$

Degree 12: $(C_3\times C_3):C_4$

Low degree siblings

24T63

Siblings are shown 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	$2^{12}$	$1$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)$
3A	$3^{4},1^{12}$	$4$	$3$	$8$	$( 3,12,19)( 4,11,20)( 7,24,15)( 8,23,16)$
3B	$3^{8}$	$4$	$3$	$16$	$( 1,17,10)( 2,18, 9)( 3,19,12)( 4,20,11)( 5,13,21)( 6,14,22)( 7,15,24)( 8,16,23)$
4A1	$4^{6}$	$9$	$4$	$18$	$( 1, 6, 2, 5)( 3,15, 4,16)( 7,20, 8,19)( 9,13,10,14)(11,23,12,24)(17,22,18,21)$
4A-1	$4^{6}$	$9$	$4$	$18$	$( 1, 5, 2, 6)( 3,16, 4,15)( 7,19, 8,20)( 9,14,10,13)(11,24,12,23)(17,21,18,22)$
6A	$6^{2},2^{6}$	$4$	$6$	$16$	$( 1, 2)( 3,20,12, 4,19,11)( 5, 6)( 7,16,24, 8,15,23)( 9,10)(13,14)(17,18)(21,22)$
6B	$6^{4}$	$4$	$6$	$20$	$( 1, 9,17, 2,10,18)( 3,11,19, 4,12,20)( 5,22,13, 6,21,14)( 7,23,15, 8,24,16)$
8A1	$8^{3}$	$9$	$8$	$21$	$( 1,16, 6, 3, 2,15, 5, 4)( 7,21,20,17, 8,22,19,18)( 9,24,13,11,10,23,14,12)$
8A-1	$8^{3}$	$9$	$8$	$21$	$( 1, 4, 5,15, 2, 3, 6,16)( 7,18,19,22, 8,17,20,21)( 9,12,14,23,10,11,13,24)$
8A3	$8^{3}$	$9$	$8$	$21$	$( 1, 3, 5,16, 2, 4, 6,15)( 7,17,19,21, 8,18,20,22)( 9,11,14,24,10,12,13,23)$
8A-3	$8^{3}$	$9$	$8$	$21$	$( 1,15, 6, 4, 2,16, 5, 3)( 7,22,20,18, 8,21,19,17)( 9,23,13,12,10,24,14,11)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	3A	3B	4A1	4A-1	6A	6B	8A1	8A-1	8A3	8A-3
	Size	1	1	4	4	9	9	4	4	9	9	9	9
	2 P	1A	1A	3A	3B	2A	2A	3A	3B	4A1	4A-1	4A-1	4A1
	3 P	1A	2A	1A	1A	4A-1	4A1	2A	2A	8A3	8A-3	8A1	8A-1
	Type
72.19.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.19.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
72.19.1c1	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$-i$	$i$	$i$
72.19.1c2	C	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$i$	$-i$	$-i$
72.19.1d1	C	$1$	$-1$	$1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-1$	$-1$	$-{\zeta }_8$	${\zeta }_8$	${\zeta }_8^3$	$-{\zeta }_8^3$
72.19.1d2	C	$1$	$-1$	$1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-1$	$-1$	${\zeta }_8^3$	$-{\zeta }_8^3$	$-{\zeta }_8$	${\zeta }_8$
72.19.1d3	C	$1$	$-1$	$1$	$1$	$-{\zeta }_8^2$	${\zeta }_8^2$	$-1$	$-1$	${\zeta }_8$	$-{\zeta }_8$	$-{\zeta }_8^3$	${\zeta }_8^3$
72.19.1d4	C	$1$	$-1$	$1$	$1$	${\zeta }_8^2$	$-{\zeta }_8^2$	$-1$	$-1$	$-{\zeta }_8^3$	${\zeta }_8^3$	${\zeta }_8$	$-{\zeta }_8$
72.19.4a	R	$4$	$4$	$-2$	$1$	$0$	$0$	$-2$	$1$	$0$	$0$	$0$	$0$
72.19.4b	R	$4$	$4$	$1$	$-2$	$0$	$0$	$1$	$-2$	$0$	$0$	$0$	$0$
72.19.4c	S	$4$	$-4$	$-2$	$1$	$0$	$0$	$2$	$-1$	$0$	$0$	$0$	$0$
72.19.4d	S	$4$	$-4$	$1$	$-2$	$0$	$0$	$-1$	$2$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed